Optimization and Realizability Problemsfor Convex Geometries

Thesis presented by Keno MERCKX

In fulfillment of the requirements for the degree of Doctor of Philosophy(Docteur en Sciences)Academic year 2018-2019

Supervisor: Jean CARDINAL

Co-supervisor: Jean-Paul DOIGNON

Samuel FIORINI (Universite libre de Bruxelles, Chair)Gwenael JORET (Universite libre de Bruxelles, Secretary)Stefan LANGERMAN (Universite libre de Bruxelles)Michel HABIB (Universite Paris Diderot)Maurice QUEYRANNE (University of British Columbia)

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UNIVERSITE LIBRE DE BRUXELLES

DOCTORAL THESIS

Optimization and RealizabilityProblems for Convex Geometries

Author:Keno MERCKX

Supervisor:Jean CARDINAL

Co-supervisor:Jean-Paul DOIGNON

Algorithms Research GroupDepartement d’Informatique

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“Pluralitas non est ponenda sine necessitate.”

William of Ockham

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UNIVERSITE LIBRE DE BRUXELLES

Faculte des SciencesDepartement d’Informatique

AbstractOptimization and Realizability Problems for Convex Geometries

by Keno MERCKX

Convex geometries are combinatorial structures; they capture in an ab-stract way the essential features of convexity in Euclidean space, graphs orposets for instance. A convex geometry consists of a finite ground set plus acollection of subsets, called the convex sets and satisfying certain axioms. Inthis work, we study two natural problems on convex geometries. First, weconsider the maximum-weight convex set problem. After proving a hard-ness result for the problem, we study a special family of convex geometriesbuilt on split graphs. We show that the convex sets of such a convex geom-etry relate to poset convex geometries constructed from the split graph. Wediscuss a few consequences, obtaining a simple polynomial-time algorithmto solve the problem on split graphs. Next, we generalize those results anddesign the first polynomial-time algorithm for the maximum-weight convexset problem in chordal graphs. Second, we consider the realizability problem.We show that deciding if a given convex geometry (encoded by its copoints)results from a point set in the plane is ∃R-hard. We complete our text with abrief discussion of potential further work.

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AcknowledgementsFirst and foremost, I would like to thank Jean Cardinal and Jean-Paul

Doignon for having accepted to supervise this thesis, and for their constantsupport during these six years. I am also very grateful for the freedom youallowed me on how I chose to organize my work. Thank you for guiding methrough this journey and for all the feedback on this document.

During my time as a student at ULB, I had the chance to meet excellentteachers. They deeply changed my vision of mathematics and computer sci-ence. Obviously Jean Cardinal and Jean-Paul Doignon, but also Samuel Fior-ini with his lectures on approximation algorithms, Gwenael Joret by sharinghis love for graph theory, and Olivier Markowitch and Yves Roggeman withtheir lectures on cryptography. I am also deeply thankful to Stefan Langer-man for his computation geometry class, and without whom I could not havestarted this work.

My gratitude goes to Michel Habib and Maurice Queyranne (in additionto Jean Cardinal, Jean-Paul Doignon, Samuel Fiorini, Stefan Langerman, andGwenael Joret who have already been cited), for accepting to be part of thejury and for reviewing my thesis.

Many thanks are also due to my friend Francois, with whom it was al-ways a pleasure to “work” with and share our happiness and frustration. Iwould like to say thank you to Udo for his interest in my work and his help-ful and challenging collaboration. I am grateful to Jeremie, Julie, Thibaut,Matthieu, Patrick, Christine and Hoan-Phung for making this experienceamazing. I would like to thank the colleagues who worked as partners indifferent teaching activities. I would also like to thank my former classmatesIsabelle, Maxime, and Rachel from ULB, and Michael, Coralie and Antoinefrom ARW. A huge thanks to Lucien, Olivier, Denis, Loraine, Laura, Lionel,Eiman and Celine.

It goes without saying that I am greatly indebted to my parents as wellas to my sister for their continuous support, encouragement, and love. Last,but not least, a heartfelt thank you goes to Marlene for all your love, supportand patience when I was only thinking about strange drawings.

I would like to finish the acknowledgments by thanking any people thatI may have forgotten to mention for their support and advice throughout theyears.

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Contents

Abstract v

Acknowledgements vii

1 Introduction: summary and main results 11.1 Convex geometries and antimatroids . . . . . . . . . . . . . . . 11.2 Finding maximum-weight convex/feasible sets . . . . . . . . . 21.3 The realizability problem . . . . . . . . . . . . . . . . . . . . . . 31.4 Contributions and collaborations . . . . . . . . . . . . . . . . . 4

2 Background 52.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Complexity theory . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 An abstract notion of convexity 93.1 Convex geometries . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 From closure operators to convex geometries . . . . . . 93.1.2 Antimatroids and shellings . . . . . . . . . . . . . . . . 123.1.3 Copoints and bases . . . . . . . . . . . . . . . . . . . . . 143.1.4 Free sets, circuits and roots . . . . . . . . . . . . . . . . 153.1.5 Occurrences, applications and research topics . . . . . 17

3.2 Examples of convex geometries . . . . . . . . . . . . . . . . . . 193.2.1 Convex geometries on posets . . . . . . . . . . . . . . . 193.2.2 Shellings of chordal graphs . . . . . . . . . . . . . . . . 203.2.3 Affine convex geometries . . . . . . . . . . . . . . . . . 223.2.4 Search antimatroids in (directed) graphs . . . . . . . . 243.2.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . 25

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4 The maximum-weight convex set problem 274.1 A classic optimization problem . . . . . . . . . . . . . . . . . . 27

4.1.1 Problem definition . . . . . . . . . . . . . . . . . . . . . 284.1.2 Computational hardness . . . . . . . . . . . . . . . . . . 294.1.3 Note on polyhedral results . . . . . . . . . . . . . . . . 31

4.2 Special cases solvable in polynomial time . . . . . . . . . . . . 314.2.1 Result for poset convex geometries . . . . . . . . . . . . 314.2.2 Result for double poset convex geometries . . . . . . . 334.2.3 Result for tree convex geometries on vertices . . . . . . 344.2.4 Result for tree convex geometries on edges . . . . . . . 354.2.5 Result for affine convex geometries in the plane . . . . 35

4.3 The case of split graphs . . . . . . . . . . . . . . . . . . . . . . . 364.3.1 Characterization of the feasible sets . . . . . . . . . . . 364.3.2 Connection between split graph shellings and posets . 414.3.3 The base poset . . . . . . . . . . . . . . . . . . . . . . . 444.3.4 Optimization results . . . . . . . . . . . . . . . . . . . . 464.3.5 Free sets and circuits characterization . . . . . . . . . . 474.3.6 Beyond this special case . . . . . . . . . . . . . . . . . . 49

5 Finding a maximum-weight convex set in a chordal graph 515.1 More on chordal graphs . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1.2 The clique-separator graph . . . . . . . . . . . . . . . . 52

5.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2.1 Main problem . . . . . . . . . . . . . . . . . . . . . . . . 535.2.2 Dummy vertices and sub-problems . . . . . . . . . . . 54

5.3 A special case solvable in polynomial time . . . . . . . . . . . 555.3.1 The rooted poset . . . . . . . . . . . . . . . . . . . . . . 565.3.2 Reduction to a poset problem . . . . . . . . . . . . . . . 58

5.4 A polynomial-time algorithm . . . . . . . . . . . . . . . . . . . 605.4.1 Computation phase . . . . . . . . . . . . . . . . . . . . . 615.4.2 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . 63

5.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.5.1 Time complexity . . . . . . . . . . . . . . . . . . . . . . 655.5.2 Detailed example . . . . . . . . . . . . . . . . . . . . . . 67

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6 The realizability problem for convex geometries 716.1 Basics of computational geometry . . . . . . . . . . . . . . . . 71

6.1.1 Affine convex geometries . . . . . . . . . . . . . . . . . 726.1.2 Abstract order types and chirotopes . . . . . . . . . . . 736.1.3 Existential theory of the reals . . . . . . . . . . . . . . . 76

6.2 Hardness result for the realizability problem . . . . . . . . . . 776.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2.2 Technical properties . . . . . . . . . . . . . . . . . . . . 786.2.3 The fixed ring . . . . . . . . . . . . . . . . . . . . . . . . 836.2.4 The reduction . . . . . . . . . . . . . . . . . . . . . . . . 84

7 Conclusion 897.1 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.2 Closing remark . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

A Allowable sequences and realizability problems 91A.1 Allowable sequences . . . . . . . . . . . . . . . . . . . . . . . . 91A.2 Realizability for allowable sequence . . . . . . . . . . . . . . . 94A.3 Results for simple allowable sequences . . . . . . . . . . . . . 95A.4 Realizability for convex geometries . . . . . . . . . . . . . . . . 96

Bibliography 101

Index 113

1

Chapter 1

Introduction: summary and mainresults

“Well, it’s rather difficult to define. Perhaps I’m just projecting my own concernabout it. I know I’ve never completely freed myself of the suspicion that there aresome extremely odd things about this mission. I’m sure you’ll agree there’s sometruth in what I say.”

— HAL 9000 in 2001: A Space Odyssey

We consider two natural problems on convex geometries: the maximum-weight convex set problem and the realizability problem. The main goal ofthis work is to obtain computational results for those questions on specificfamilies of convex geometries. The theorems presented in this section are themain original results of this thesis. The notions used here will be formallyintroduced later on with references and detailed examples.

1.1 Convex geometries and antimatroids

Chapter 3 of this work presents the concept of convex geometries, togetherwith some examples. Roughly said, a convex geometry is defined by a finiteset, called the ground set, and some special subsets of this ground set thatwe call “convex sets”. These special subsets must share some well-definedproperties. Formally, a set system (V, C) is a convex geometry if ∅ ∈ C, theset C is stable under intersection, and for all C in C \ V, there exists a c inV \C such that C∪c is also in C. We also study structures related to convexgeometries: antimatroids. Formally, a set system (V,F ) is an antimatroidif V ∈ F , the set F is stable under union, and for all F in F \ ∅, thereexists an f in F such that F \ f is also in F . The sets in F are called thefeasible sets. Convex geometries are related to antimatroids in the following

2 Chapter 1. Introduction: summary and main results

sense: (V, C) is a convex geometry if and only if (V, V \ C : C ∈ C) is anantimatroid.

1.2 Finding maximum-weight convex/feasible sets

Chapters 4 and 5 concern the maximum-weight convex/feasible set problem.For a given antimatroid with some real weight on each element of its groundset, a natural problem is finding a feasible set with maximum weight, wherethe weight of a feasible set is the sum of the weight of its elements. Of course,the problem in a convex geometry is closely related to the maximum-weightconvex set problem. The antimatroids are encoded by their bases i.e. thefeasible sets that cannot be obtained as the union of two other feasible sets.

In Chapter 4, we adapt a result of Eppstein and show that the maximum-weight feasible set problem is hard.

Theorem. The problem of finding a maximum-weight feasible set in an antimatroidencoded in the form of its base poset is not approximable in polynomial time withina factor better than O(N

12−ε) for any ε > 0, where N is the number of elements in

the base poset, unless P = NP .

The proof relies on a reduction from the problem of finding the maximumsize of an independent set in a graph.

Next, we study the maximum-weight feasible set problem restricted toantimatroids built on split graphs (i.e. graphs in which the vertices can bepartitioned into a clique K and an independent set I). Given a split graphand using the set of vertices as ground set, the feasible sets of the split graphshelling antimatroid are by definition, the prefixes of perfect elimination or-derings (i.e. iteratively removing a vertex whose neighborhood induces aclique). We show the following polynomial time result.

Theorem. Given a split graph G = (K ∪ I, E), a maximum-weight feasible set inthe split graph shelling antimatroid defined on G can be found in polynomial time.

We design an algorithm to solve the problem with time complexity inO(|I|TMClo(|K| + |I|, |E|)) where TMClo(n, m) is the time complexity forsolving the maximum-weight ideal problem for a poset with n elements andm cover relations. We also describe a new characterization of free sets (thefeasible sets in which every subset is also feasible) and circuits (the minimal

1.3. The realizability problem 3

non-free sets) for the split graph shelling antimatroid. The main idea behindthese results is to define partially ordered sets on the vertices of the splitgraph and use algorithmic and structural properties of the resulting partiallyordered sets.

In Chapter 5 we tackle the the maximum-weight convex set problem inchordal graphs (i.e. graphs with no induced cycle of length four or more).For a chordal graph, we use the notion of m-convex sets defined as follow:a set of vertices is a m-convex set if it contains the vertices of all chordlesspaths between any two vertices of the set. All m-convex sets form a convexgeometry: it is called the monophonically convex geometry. We obtain thefollowing theorem which generalizes the previous result.

Theorem. Given a chordal graph G = (V, E), a maximum-weight convex set inthe convex geometry defined on G (by the m-convex sets) can be found in polynomialtime.

The time complexity of the algorithm is O(|V|2|E|2). Note that the algo-rithm for solving the problem on split graphs has a better time complexity,but cannot be applied to all chordal graphs. The idea of the proof is to useseparators (i.e. a subset of vertices such that, when removed, the number ofconnected components increases) in the graph to break the problem downinto a collection of simpler subproblems on partially ordered sets.

1.3 The realizability problem

In Chapter 6 we look at a realizability problem. In the plane, a finite point setP naturally induces an (affine) convex geometry defined by the family of con-vex sets C ⊆ 2P : conv(C) ∩ P = C. We say that a convex geometry (V, C)is realizable if there exists a point set V in the plane such that the convex ge-ometry induced by V is isomorphic to (V, C). The realizability problem forconvex geometries is, given a convex geometry (V, C), deciding if (V, C) isrealizable. Here the convex geometry is encoded by a set of copoints (i.e. theconvex sets that cannot be obtained as the intersection of two other convexsets). We prove the following theorem.

Theorem. The realizability problem for convex geometries encoded by a copointposet is ∃R-complete.

4 Chapter 1. Introduction: summary and main results

Here, the existential theory of the reals (∃R) is a complexity class definedby the following complete problem: given a system of polynomial equali-ties and inequalities (with real coefficients), decide if there is an assignmentof real values to the variables such that the system is satisfied. Moreover,we show that the realizability problem is ∃R-hard even for a specific fam-ily of convex geometries defined on abstract order types (which are a strictabstraction of point sets also known as acyclic oriented matroids of rank 3).The main idea is to use a the hardness of the realizability problem for abstractorder types and set up a polynomial reduction.

1.4 Contributions and collaborations

Parts of this dissertation have been or are to be published in internationalpeer-reviewed journals. Results in Chapters 4 and 5 were obtained in collab-oration with Jean Cardinal and Jean-Paul Doignon, they have been publishedin Discrete Mathematics & Theoretical Computer Science [25] and Journal of GraphAlgorithms and Applications [24] respectively. Results in Chapters 6 were ob-tained in collaboration with Udo Hoffmann, they are currently under reviewfor publication in Journal of Computational Geometry [67]. The results obtainedin collaboration with Francois Gerard and published in Cryptology and Net-work Security [53] are not part of this work.

5

Chapter 2

Background

“It’s always difficult to keep personal prejudice out of a thing like this. Andwherever you run into it, prejudice always obscures the truth.”

— Juror #8 in 12 Angry Men

We review here some basic definitions and notation used in this work.

2.1 Basics

We use the classical notation from set theory. The set 2V consists of the sub-sets of the set V. The cardinality of V is noted |V|. The symbol ∅ refers to theempty set. The symbol R refers to the set of real numbers. The complemen-tary set V \ X of X for X ⊆ V is denoted by X. A pair (V,X ), where V is afinite set of elements called the ground set andX ⊆ 2V , is called a set system.For f and g real-valued functions, both defined on some unbounded subsetof the real positive numbers, such that g(x) is strictly positive for all largeenough values of x, we write f (x) 6 O(g(x)) if and only if for all sufficientlylarge values of x, the absolute value of f (x) is at most a positive constantmultiple of g(x). For a finite set V, the function w is a weight function ifw : V → R. For a weight function w on a set V, we extend the functions to2V as follow: for U ⊆ V, the real value of w(U) is the sum of w(u) for all u inU. For all classical set theory formalism see Bourbaki [20].

2.2 Graph theory

A (simple) graph G is a pair (V, E) where V is a (finite) set of vertices and Ea set of unordered pairs of vertices, the elements in E are called the edges. Adirected graph G is a pair (V, A) where V is a (finite) set of vertices and A

6 Chapter 2. Background

a set of ordered pairs of vertices, the elements in A are called the arcs. In agraph G = (V, E) (resp. a directed graph G = (V, A)), a path is a sequenceof distinct vertices (v1, . . . , vn) such that vi, vi+1 ∈ E (resp. (vi, vi+1) ∈ A)for all i in 1, . . . , n− 1. A path is chordless if no two vertices are connectedby an edge that is not in the path. The graph G′ = (V′, E′) is a subgraph ofG = (V, E) if V′ ⊆ V and E′ ⊆ E. The subgraph G′ = (V′, E′) of G = (V, E)is induced if u, v ∈ V′ and u, v ∈ E imply u, v ∈ E′. The graph is con-nected if for any u, v in V there is a path (u, . . . , v). A connected componentof G is a maximal connected subgraph of G. Each vertex belongs to exactlyone connected component, as does each edge. In a graph G = (V, E) (resp.a directed graph G = (V, A)), a cycle is a path (v1, . . . , vn) such that vn, v1is an edge (resp. (vn, v1) is an arc). A cycle is chordless if no two verticesof the cycle are connected by an edge that does not itself belong to the cycle.A graph is chordal if every chordless cycle in the graph has at most threevertices. For V′ ⊆ V we denote by N(V′) the set of vertices w in V \V′ suchthat w, v ∈ E for some v in V′. We write N(v) for N(v). For U ⊆ V wedefine G−U as the graph induced by V \U, we write G− v for G−vwithv in V. A clique K of G is a set of pairwise adjacent vertices, we say that K isa maximal clique if there is no clique K′ of G such that K ⊂ K′. We denote byKG the set of all maximal cliques in G. We call a vertex v simplicial if N(v)induces a clique. A set I of vertices is independent if no two vertices in I areadjacent. In a directed graph, a source is a vertex without incoming arcs anda sink is a vertex without outgoing arc. A directed graph G is an s, t-graph ifG has no cycle, only one source (called s) and only one sink (called t). For abackground on graph theory we recommend Diestel [33].

2.3 Order theory

A partially ordered set (or poset) P is a pair (V,6) formed of a finite set Vand a binary relation 6 over V which is reflexive, antisymmetric, and transi-tive. For a poset (V,6) an ideal I is a subset of V such that for all elementsa in I and b in V, if b 6 a, then b is also in I. The ideals are also known asdownsets. We call idl(P) the set of ideals in P. A filter F is a subset of V suchthat for all elements a in F and b in V, if a 6 b, then b is also in F. The filtersare also known as upsets. We denote the family of all filters of P by flt(P).For u, v in V, we say that v covers u in P if u 6= v, u 6 v and there is no x

2.4. Geometry 7

in V \ u, v such that u 6 x 6 v. We usually use the Hasse diagram of aposet to represent it: concretely, for a partially ordered set (V,6) we repre-sent each element of V as a point in the plane and draw a line segment orcurve that goes upward from u to v whenever v covers u. These curves maycross each other but must not touch any points other than their endpoints.Such a diagram, with labeled points, uniquely determines its partial order.We recommend Trotter [119] for more details on poset theory.

2.4 Geometry

A subset K of Rn is a Euclidean convex set if for every x, y in K, the segment[x, y] defined by λx + (1− λ)y : 0 6 λ 6 1 is contained in K. For X inRn, the convex hull of X, noted conv(X) is the (unique) smallest Euclideanconvex set (with respect to inclusion) that contains X. A point set P in R2 is ingeneral position if, for all line ` spanned by p1 and p2 in P, we have that P ∩` = p1, p2. For a finite set E, the pair (E, χ) is an abstract order type if χ :E3 → −1, 0, 1 satisfies the following properties. First, χ is not identicallyzero. Second, χ(pρ(1), pρ(2), pρ(3)) = sgn(ρ)χ(p1, p2, p3) for p1, p2, p3 ∈ E andany permutation ρ. Third, there are p1 and p2 in E such that χ(p1, p2, q1) > 0for all q1 ∈ E. Fourth, if χ(q1, p2, p3)χ(p1, q2, q3), χ(q2, p2, p3)χ(q1, p1, q3) andχ(q3, p2, p3)χ(q1, q2, p1) are not −1 then χ(p1, p2, p3)χ(q1, q2, q3) is also not−1, for any p1, p2, p3, q1, q2, q3 in E. The notion of abstract order type is ageneralization of the orientation (clockwise, collinear, or counterclockwise)of every ordered triple of points in a finite subset E of the plane R2. Notethat abstract order types are also called acyclic oriented matroids of rank 3.For more details on convexity, abstract order type and oriented matroids seeGoodman et al. [55].

2.5 Complexity theory

A complexity class is a set of problems of related computational complexity.We denote by P the class of decision problems solvable in polynomial timeby a deterministic Turing machine. The class NP is the class of decisionproblems solvable by a nondeterministic polynomial-time Turing machine.An algorithm is a factor α approximation for a problem if and only if forevery instance of the problem it can find a solution within a factor α of the

8 Chapter 2. Background

optimum solution. If the problem is a minimization then α > 1 and thedefinition implies that the solution found by the algorithm is at most α timesthe optimum solution. On the other hand, for maximization problem, wehave α < 1 so the definition guarantees that the approximate solution is atleast α times the optimum. See Sipser [116] for more details on computationalcomplexity.

9

Chapter 3

An abstract notion of convexity

“Part of me was afraid of what I would find and what I would do when I gotthere. I knew the risks, or imagined I knew. But the thing I felt the most, muchstronger than fear, was the desire to confront him.”

— Capt. Benjamin Willard in Apocalypse Now

The goal of this chapter is to introduce the notions of convex geometryand antimatroid along with the complementarity principle between those ob-jects. After defining the basic concepts, we recall some structural results. Asthose notions were frequently “rediscovered”, we dedicate a section to por-tray the use of convex geometries and antimatroids in different branches ofmathematics. This will be followed by a list of examples and the descriptionof different families of convex geometries used in the literature. This chapteris mainly based on Korte et al. [83], and the paper of Edelman and Jami-son [42]. We also use some notions from matroid theory, see Oxley [99] formore details. Proofs omitted in this chapter can be found in those references.

3.1 Convex geometries

In this section, we see how an abstraction of Euclidean convex sets leads tothe concept of convex geometry.

3.1.1 From closure operators to convex geometries

Many approaches to convex geometries and antimatroids are described in theliterature, see for instance Monjardet [94]. We give the one based on closureoperators because it highlights the link between antimatroids and matroids.Let V be a finite set and C a collection of subsets of V with the following

10 Chapter 3. An abstract notion of convexity

properties:

∅ ∈ C and V ∈ C,

C1, C2 ∈ C ⇒ C1 ∩ C2 ∈ C.

The set C is called an alignment of V. This leads to the operator over 2V :

τ(X) =⋂C ∈ C : X ⊆ C,

where X is a subset of V. We observe that τ(X) is the unique minimal set (forthe inclusion) in C that contains X. Moreover, τ is a closure operator as it hasthe following properties:

X ⊆ τ(X),

X ⊆ Y ⇒ τ(X) ⊆ τ(Y),

τ(τ(X)) = τ(X),

τ(∅) = ∅.

Note that we have the following equality for a finite set V and a closureoperator τ:

τ(X) : X ∈ 2V = X ∈ 2V : X = τ(X).

The set in both sides of the equality describes the elements in the alignment C,these will be called the closed sets. We call the set V together with the closureoperator τ a convex geometry if, for all X ⊆ V the following property, calledthe anti-exchange property, is verified.

If y, z /∈ τ(X), y 6= z and z ∈ τ(X ∪ y), then y /∈ τ(X ∪ z).

Note that the anti-exchange property is verified in the case of the Eu-clidean convex hull. Indeed, if we have two distinct points y and z outsideof a convex hull of a set X and if z is in conv(X ∪ y), then y is outside ofconv(X ∪ y). This is illustrated by Figure 3.1.

For convex geometries, the closed sets are called the convex sets. In prac-tice, we define a convex geometry by a pair (V, C), where V is a finite set of

3.1. Convex geometries 11

z

y

conv(X)

FIGURE 3.1: Illustration of the anti-exchange property for theconvex hull in the plane.

elements and C ⊆ 2V satisfies

∅ ∈ C,

∀C1, C2 ∈ C : C1 ∩ C2 ∈ C,

∀C ∈ C \ V, ∃ c ∈ V \ C : C ∪ c ∈ C.

The convex sets are the elements in C. This definition is equivalent to the onewith the closure operator, as stated by Korte et al. [83]. Figure 3.2 gives twoexamples of convex geometries on the ground set 1, 2, 3, 4, whose collec-tion of convex sets is given as a poset with the cover relation.

∅

1 2

1, 2

1, 2, 3

1, 2, 3, 4

∅

1 2 3

1, 3 2, 3 3, 4

1, 2, 3 1, 3, 4 2, 3, 4

1, 2, 3, 4

FIGURE 3.2: Two examples of convex geometries on four ele-ments.

We note here that the name convex geometry is controversial: it names amathematical object as well as an entire discipline. Unfortunately, it has be-come more or less standard in the field. If we look back at the anti-exchangeproperty, we see a link with the exchange property which requires for a clo-sure operator σ : 2V → 2V that for all X ⊆ V and for all y, z in V we have:

If z /∈ σ(X) and z ∈ σ(X ∪ y), then y ∈ σ(X ∪ z).

12 Chapter 3. An abstract notion of convexity

The exchange property is the same as the MacLane–Steinitz exchangeproperty used in the standard matroid definition, see MacLane [87] and Ox-ley [99] for details. Note that the anti-exchange property is a fundamentalfeature of the convex hull in a real affine space, while the exchange propertyis a fundamental feature of the affine hull.

To stay consistent with the notation in the Euclidean case, we call an ele-ment c of a convex set C in a convex geometry defined by τ an extreme pointof C if c /∈ τ(C \ c). The set of extreme points of C is noted ext(C). Thenext theorem gives two interesting characterizations of convex geometries,proofs are given in Edelman and Jamison [42].

Theorem 3.1.1. For a finite set V, a closure operator τ : 2V → 2V and C = τ(X) :X ⊆ V, the three statements are equivalent:

(i) the pair (V, C) is a convex geometry;

(ii) if C ( V is a convex set, there exists x in V \ C such that C ∪ x is convex;

(iii) for all convex sets C in V, we have C = τ(ext(C)).

3.1.2 Antimatroids and shellings

We will now define structures which are, in a sense, duals of convex geome-tries: antimatroids. We also detail below the link between the two notions.Let V be a finite set and let F be a non-empty family of subsets of V. We saythat (V,F ) is an antimatroid if

V ∈ F ,

F1, F2 ∈ F ⇒ F1 ∪ F2 ∈ F ,

F ∈ F \ ∅ ⇒ ∃ f ∈ F : F \ f ∈ F .

The last condition above is often called the accessibility property. Thename antimatroid comes from the fact that these objects are closely related tothe anti-exchange property, the opposite, in a way, of the exchange propertyfor matroids. Note that many authors find the name barbaric and poorlyadapted. The elements of F will be called the feasible sets. Because thefeasible sets are closed under union, the set of their complements F = V \F : F ∈ F is closed under intersection, which gives us a closure operator.

3.1. Convex geometries 13

From this, we reach the following proposition, a proof can be found in Korteet al. [83].

Proposition 3.1.1. The pair (V,F ) is an antimatroid if and only if (V, C) is aconvex geometry with C = V \ F : F ∈ F.

1, 2, 3, 4

2, 3, 4 1, 3, 4

3, 4

4

∅

1, 2, 3, 4

2, 3, 4 1, 3, 4 1, 2, 4

2, 4 1, 4 1, 2

4 2 1

∅

FIGURE 3.3: Two examples of antimatroids on four elements.

Figure 3.3 shows two antimatroids on the ground set 1, 2, 3, 4, the fea-sible sets are given as a poset with the cover relation. We easily see that thecomplement of those feasible sets are the ones illustrated by Figure 3.2. Weuse the above proposition to adapt the definitions of feasible and convex setsas follows. For a convex geometry (V, C) the elements of V \ C : C ∈ Care the feasible sets. Similarly, for an antimatroid (V,F ), the element ofV \ F : F ∈ F are the convex sets. Because of the complementarity ex-posed in Proposition 3.1.1, every result on convex geometries can be trans-lated into a result about antimatroids and vice versa. In this work we willuse both points of view depending on the context and the results we want toshow.

Antimatroids also relate to special elimination processes. An eliminationprocess of a set V designates a general procedure in which the elements of Vare removed one at a time according to a given rule. If every element whichis removable at some stage of the process remains removable at any laterstage, we call this a shelling process. Given a feasible set F in an antimatroid(V,F ), a shelling of F is an enumeration f1, f2, . . . , f|F| of its elements suchthat f1, f2, . . . , fk is feasible for any k with 1 6 k 6 |F|. The underlyingrules here is: the element f in V is removable at step i if Fi−1 ∪ f is in Fwhere Fi−1 is the set of elements removed at steps 1 to i− 1. This is indeed ashelling process. In view of the accessibility property, any feasible set admitsat least one shelling. There is an axiomatization of antimatroids in terms of

14 Chapter 3. An abstract notion of convexity

shelling, for details see Korte et al. [83]. Many examples of antimatroids arisein a natural way from shelling processes as shown by Goecke et al. [54]. Someauthors such as Eppstein [45] call basic words of an antimatroid (V,F ) anyshelling of the whole set V.

3.1.3 Copoints and bases

Let (V, C) be a set system and C in C. An element f in V \ C is attachedat C if C ∪ f is also in C. For an element v in V, the set C is a copointif C is a maximal set (with respect to inclusion) of C contained in V \ v.There may be more than one copoint attached at an element. For convexgeometries, this notion of copoint allows another characterization as statedby in Edelman and Jamison [42].

Theorem 3.1.2. The pair (V, C) with C an alignment of V is a convex geometry ifand only if every copoint C has a unique attaching element.

For a convex geometry, the copoints are also the convex sets that cannotbe obtained as the intersection of two other convex sets, as stated by thefollowing proposition from Korte et al. [83]. Figure 3.4 shows the list of allcopoints for a given convex geometry.

Proposition 3.1.2. Let (V, C) be a convex geometry and C a convex set with kelements attached at it. Then C is the intersection of k copoints.

∅

1 2 3

1, 3 2, 3 3, 4

1, 2, 3 1, 3, 4 2, 3, 4

1, 2, 3, 4Copoints Attachment1 32 31, 2, 3 41, 3, 4 22, 3, 4 1

FIGURE 3.4: Illustration of copoints in a convex geometry.

In particular, a proper subset of V is convex if and only if it is the in-tersection of copoints. Because the convex sets of a convex geometry (V, C)are closed under intersection, a natural way to encode (V, C) is to only keep

3.1. Convex geometries 15

track of the copoints. It is very common to order the copoints with the inclu-sion relation to obtain the copoint poset. An illustration of a copoint poset isgiven in Figure 3.5.

∅

1 2 3

1, 3 2, 3 3, 4

1, 2, 3 1, 3, 4 2, 3, 4

1, 2, 3, 4

1 2

1, 2, 3 1, 3, 4 2, 3, 4

FIGURE 3.5: A convex geometry and its copoint poset.

Because of the duality between convex and feasible sets, the concept ofcopoints and attachment are also used in antimatroids. This leads us to thefollowing definitions regarding the feasible sets of a convex geometry. Abase of a convex geometry is a non-empty feasible set that is not the unionof two other feasible sets. Note that the notion of base often appears underthe name “path”, but the term becomes confusing when dealing with graphtheory where the notion of path is used for something completely different.In the same way we have defined the copoint poset, it is easy to obtain thebase poset by ordering the bases with the inclusion relation. We use the termendpoint of a feasible set F to denote an element f in F such that F \ f isalso feasible. Obviously, a base has a unique endpoint.

3.1.4 Free sets, circuits and roots

We will now focus on the notion of free sets, circuits and roots. Those con-cepts are designed to fit the antimatroid point of view. Let (V, C) be a convexgeometry with F the set of its feasible sets. For X ⊆ V, we define the traceof (V, C) on X as

Tr(F , X) = F ∩ X : F ∈ F.

Using the definition of an antimatroid given in Subsection 3.1.2, we see thatTr(F , X) is also an antimatroid. We say that a set X ⊆ V is free if Tr(F , X) =

2X. We call circuit a set T ⊆ V if it is minimal non-free, in other words, if T

16 Chapter 3. An abstract notion of convexity

is not free but every proper subset of T is free. Next, we define for a feasibleset F the feasible continuation of F as

Γ(F) := c ∈ V \ F : F ∪ c ∈ F.

From a convex geometry point of view, Γ(F) is the set of extreme points of thecomplement of F in V. In other words Γ(F) = ext(V \ F). The circuits playan important role in the study of convex geometry and antimatroids becausethey lead to another nice characterization. We call the interior of a set X ⊆ Vthe feasible set F such that F ⊆ X and F is maximal for this property. Fromthe union stability of the feasible sets, the interior of a set is unique. The nextproposition will help us to define the concept of root, again the proof is inKorte et al. [83].

Proposition 3.1.3. Let (V, C) be a convex geometry with F the set of its feasiblesets, a circuit T and I the interior of V \ T. Then Γ(I) ⊆ T and |T \ Γ(I)| = 1.

Now, for a circuit T we call the root of T the unique element r in T suchthat r = T \ Γ(I), where I is the interior of V \ T. The set T \ r will becalled the stem. Usually we write (T \ r, r) to denote a circuit T with root r.Figure 3.6 gives an illustration of free sets and circuits in a convex geometry.The next proposition gives a more intuitive view of the circuits.

Proposition 3.1.4. Let (V, C) be a convex geometry with F the set of its feasiblesets, T ⊆ V and r ∈ C. Then T is a circuit with root r if and only if Tr(F , T) =

2T \ r.

We close this section with a characterization of antimatroids by their cir-cuits due to Dietrich [34]. For a set system (V,F ), the author call rootedsubset a pair (X, r) such that X ⊆ V and r ∈ V \ X.

Theorem 3.1.3. Let K be a set of rooted subsets of a finite set V. Then K is theset of circuits of an antimatroid (V,F ) if and only if K satisfies the following twoproperties:

(C1, r), (C2, r) ∈ K, C1 ⊆ C2 ⇒ C1 = C2,

∀(C1, r1), (C2, r2) ∈ K, r1 ∈ C2 \ r2, ∃(C3, r2) ∈ K with C3 ⊆ C1 ∪ C2 \ r1.

3.1. Convex geometries 17

∅

1 2 3

1, 3 2, 3 3, 4

1, 2, 3 1, 3, 4 2, 3, 4

1, 2, 3, 4

C

1, 2, 3, 4

2, 3, 4 1, 3, 4 1, 2, 4

2, 4 1, 4 1, 2

4 2 1

∅

F

Free sets Circuits1241, 21, 4 (1, 4, 3)2, 4 (2, 4, 3)1, 2, 4 (1, 2, 4, 3)

FIGURE 3.6: Illustration of free sets and circuits in a convexgeometry.

3.1.5 Occurrences, applications and research topics

The concept of convex geometry has been discovered several times as ex-plained by Monjardet [94]. Dilworth [35] first examined structures very closeto convex geometries such as the class of lower locally distributive lattices.He was followed by Avann [9, 10] who studied lower semidistributive lattice,Boulaye [18, 19] with the α-weakened lattices, Pfaltz [103] and the concept ofG-lattice, Greene and Markowsky [59] with the notion of lower locally dis-tributive lattice.

Edelman [40] and Jamison [70, 71] studied in detail the convex geome-tries. Together, Edelman and Jamison [42] presented the foundations of thetheory of convex geometries. Korte et al. [83] considered antimatroids as asubclass of greedoids which have been introduced by Korte and Lovasz [81]and can be viewed as accessible set systems that satisfy the exchange prop-erty.

In choice function theory, the concept of path independence of a choicefunction was suggested by Plott [106] in order to weaken the condition of

18 Chapter 3. An abstract notion of convexity

rationality in a manner which preserves one of the key properties of ratio-nal choice, namely that choice over any subset should be independent of theway the alternatives were initially divided up to consideration. Later, Ko-shevoy [84] shows that a choice function is path independent if and only ifsome closure operator attached to it defines a convex geometry. This relationhas also been studied by Monjardet and Raderanirina [95]. More recently,this connection between choice functions and convex geometries led to somegeneralizations by Danilov and Koshevoy [32].

In mathematical psychology the concepts of learning space and antima-troid are equivalent, see Falmagne and Doignon [37, 48] for details. In thelearning space paradigm, the feasible sets represent a set of knowledge statesfrom students, and the feasible continuation represents the items this studentwill be able to learn next based on his current state. This theory has led to thedesign of ALEKS (Assessment and Learning in Knowledge Spaces), a com-puter system, and a company built around that system. This helps studentslearn knowledge-based academic systems such as mathematics by assess-ing their knowledge and providing lessons in concepts that the assessmentjudges them as ready to learn. This development opens the door for manyapplications, see Eppstein [45], Doignon et al. [38] and Yoshikawa et al. [123].

The convex geometries also appear in many other fields of mathemat-ics. We mention here the formal language theory, see Crapo [29], Boyd andFaigle [21] and Kempner and Levit [76]. We can also mention game theory,see Algaba and et al. [7] and more recently Jimenez-Losada and Ordonez [74].

There are various research questions around convex geometries. With-out being exhaustive we mention a few problems. The counting problemsfor a small numbers of elements, see Uznanski [120] or asymptotically, seeEchenique [39]. Eppstein [44] generalized to antimatroids the 1/3-2/3-con-jecture from ordered sets. There also exist some polyhedral and optimiza-tion results given, among others, by Korte and Lovasz [82], Knauer [79] andQueyranne and Wolsey [107, 108]. There are representation theorems, seefor instance Kashiwabara et al. [75] and Adaricheva and Wild [5]. Infiniteversions of convex geometries occur in several sources, see for instance Dil-worth and Crawley [36], Semenova [114], Wahl [121], Adaricheva et al. [2],Adaricheva and Nation [4] and Mao [90].

3.2. Examples of convex geometries 19

3.2 Examples of convex geometries

In this section we provide some families of convex geometries and antima-troids. We skip the known proofs that the different properties given in theprevious section applied to the structures below. The goal here is not to givean exhaustive list of all families of convex geometries, but to present someclassical examples used in the literature. We also give more context and de-tails on convex geometries which are used later in this work. We switchbetween the convex and feasible sets points of view in order to give the mostintuitive idea of the notions. Fore more examples, see for instance Goecke etal. [54].

3.2.1 Convex geometries on posets

One particular class of convex geometries comes from shelling processes overposets by removing successively maximal elements. Let (V,6) be a poset,then (V, idl(V,6)) is a poset convex geometry. Thus the convex sets arethe ideals and the feasible sets are the filters. By duality, (V, flt(V,6)) isa poset antimatroid. The class of poset convex geometries is often consid-ered as most basic ones, because it arises in many different contexts. Posetconvex geometries are the only convex geometries closed under union asshown in Korte et al. [83]. There exist several other characterizations for thisclass of convex geometries. Nakamura [97, 96] obtains a characterization ofposet convex geometries by single-element extensions and by excluded mi-nors. Recently, Kempner and Levit [77] introduced the poly-dimension ofan antimatroid, and proved that every antimatroid of poly-dimension 2 is aposet antimatroid. They established both graph and geometric characteriza-tions of such convex geometries. A shelling in a poset antimatroid coincidewith a linear extensions of the poset (which explains their relationship to the1/3-2/3-conjecture mentioned in Eppstein [44]). We also note the Represen-tation Theorem due to Birkhoff [14]: When ordered by inclusion, the feasiblesets of a poset antimatroid form a distributive lattice. Conversely, any dis-tributive lattice is isomorphic to some poset antimatroid. Circuits are of theform (v, r) with v, r in V and v < r. Moreover an antimatroid is a posetantimatroid if and only if all of its circuits have cardinality 2, as proved byKorte et al. [83]. It is easy to show that the set of copoints of (V, idl(V,6)) isV \ flt(v) : v ∈ V. Figure 3.7 gives an example of a poset convex geometry.

20 Chapter 3. An abstract notion of convexity

43

5

1 2 ∅

21

2, 41, 2

1, 2, 41, 2, 3

1, 2, 3, 4

1, 2, 3, 4, 5

FIGURE 3.7: A poset and the poset convex geometry built on it.

There is another popular notion of convexity on posets, instead of shellingfrom the top we could shell it from the bottom and the top. Let (V,6) be aposet, then (V, C) is a double poset convex geometry when the sets in Care the intersections between a filter and an ideal. By duality, (V,F ) withF = X ∪ Y : X ∈ idl(V,6), Y ∈ flt(V,6) is a double poset antimatroid.For u and v in a convex set C, we have directly that if u 6 c 6 v then c isin C. Formally introduced by Edelman [40], the circuits of the double posetantimatroids are (u, v, x) such that u < x < v and u, v, x in V. Figure 3.8gives an example of a double poset convex geometry.

32

4

1 ∅

32 41

1, 21, 3 2, 42, 3 3, 4

1, 2, 3 2, 3, 4

1, 2, 3, 4

FIGURE 3.8: A poset and the double poset convex geometrybuilt on it.

3.2.2 Shellings of chordal graphs

Chordal graphs lead to several convex geometries. We first recall some well-known examples of such graphs. A tree is a graph in which any two verticesare connected by exactly one path. In a tree, a vertex v with |N(v)| = 1 is

3.2. Examples of convex geometries 21

called a leaf. A split graph is a graph whose set of vertices can be partitionedinto a clique and an independent set. A Ptolemaic graph is a chordal graphin which the distances in any connected induced subgraph are the same asthey are in the original graph. Finally, a graph is a block graph if the inter-section of every two connected subsets of vertices of the graph is empty orconnected.

We start by looking at two types of convex geometries defined on trees.The first idea is use the vertices as ground set. For a tree T = (V, E) the pair(V, C) is a tree convex geometry on vertices when C is composed of the ver-tex sets of all connected subgraphs of T. A vertex shelling tree antimatroidis obtained by the shelling processes on vertices that consist of repeatedlydeleting leaves of the remaining tree. The circuits of the vertex shelling treeantimatroid are (u, v, r) such that r is on the path between u and v, for u,v, r distinct vertices in V.

There is another popular notion of convexity on trees, instead of definingthe shelling on vertices we could define it on the edges. For a tree T = (V, E)the pair (E, C) is a tree convex geometry on edges where C is composedof the edges sets of all connected subgraphs of T. An edge shelling treeantimatroid is obtained by the shelling processes on the set of vertices thatconsist of repeatedly deleting edges of the tree that contain a leaf. The circuitsof the edge shelling tree antimatroid are (e, f , r) such that in the graph T− rthe edges e and f are in separate connected components.

For a graph G = (V, E), a set C of vertices is a connected convex set (alsoknown as c-convex set) if the subgraph induced by C is connected. Givena graph G = (V, E) with C the set of c-convex sets of G, it happens that(V, C) is a convex geometry if and only if G is a block graph, see Jamison [72]for a proof. The convex geometry (V, C) is then called a connected convexgeometry. We can also use the shortest paths in a graph G = (V, E) to obtaina notion of convexity. A set C ⊆ V of vertices is a geodesic convex set (alsoknown as g-convex set) if C contains every vertex on every shortest pathbetween vertices in C. Given a graph G = (V, E) with C the set of g-convexsets of G, it happens that (V, C) is a convex geometry if and only if G is adisjoint union of Ptolemaic graphs, see Farber and Jamison [49] for a proof.The convex geometry (V, C) is then called a geodesic convex geometry.

For a graph G = (V, E), a set C of vertices is a monophonically convexset (also known as m-convex set) if C contains every vertex on every chord-less path between vertices in C. Given a graph G = (V, E) with C the set of

22 Chapter 3. An abstract notion of convexity

m-convex sets of G, it happens that (V, C) is a convex geometry if and onlyif G is chordal, see Farber and Jamison [49] for a proof. The convex geometry(V, C) is then called a monophonically convex geometry. The dual notioncomes by defining a shelling process that removes the simplicial vertices ofa chordal graph, leading to a chordal graph antimatroid. See Chvatal [28]for a description of monophonically convex geometries using the conceptof “betweenness”. In some works the special case of monophonically convexgeometry in split graphs is studied, for instance in Eppstein [44]. Like doubleposet antimatroids, the circuits of chordal shelling antimatroids have cardi-nality 3. The pair (u, v, r) is a circuit if and only if there is a chordless pathfrom u to v passing through r. Figure 3.9 gives us the relation of inclusion be-tween the different families defined above. Note that the convexity notions

Tree convex geometry on vertices

Tree convex geometry on edges

Connected convex geometry

Geodesic convex geometry

Monophonically convex geometry

FIGURE 3.9: The inclusion relation for different classes of con-vex geometries defined on families of graphs.

defined here on graphs can be generalized to hypergraphs. See Farber andJamison [49] for more details. Figure 3.8 gives an example of a monophoni-cally convex geometry built on a chordal graph, which is in this case also asplit graph.

3.2.3 Affine convex geometries

Let V ⊂ Rn a finite point set. We use the convex hull to define an affineconvex geometry (V, C) with C = C ⊆ 2V : conv(C) ∩ V = C. Here theunderlying shelling structure corresponds to a recursive deletion of extremepoints of the convex hull generated by the remaining points. This processcan be generalized using a cone T and allowing only shelling in the directionof T. More formally, from a finite point set V in Rn and a cone T ⊆ Rn,the pair (V, C) is a lower affine convex geometry where C = C ⊆ 2V :

3.2. Examples of convex geometries 23

1

2

3

4

∅

1 2 3 4

1, 2 2, 4 2, 3 1, 3 3, 4

1, 2, 3 2, 3, 4

1, 2, 3, 4

FIGURE 3.10: A chordal graph and the monophonically convexgeometry build on it.

(conv(C) + T) ∩ V = C. Figure 3.8 gives an example of an affine convexgeometry built on a set of points in R2.

R2

4

3

1 2 ∅

1 2 3 4

1, 2 1, 3 1, 4 2, 3 2, 4 3, 4

1, 2, 4 1, 3, 4 2, 3, 4

1, 2, 3, 4

FIGURE 3.11: A point set and the affine convex geometry buildon it.

The affine convex geometries have been generalized using abstract ordertypes and acyclic oriented matroids, see Bland [16] and Las Vergnas [86] fordefinitions and detailed constructions. Given an abstract order type (V, χ).For X ⊆ V, we define the closure operator σχ as follows, the set σχ(X) is theintersection of the sets H+ ⊆ V such that X ⊆ H+ and there are h1, h2 in H+

with χ(h1, h2, x) > 0 for all x in X. Now, given an abstract order type χ wedefine an AOT convex geometry (V, C) with C = C ⊆ 2V : σχ(C) = C. Theletters AOT is the acronym for abstract order type. Note that the sets in H+

is an abstraction of the half-spaces generally used in the Euclidean convexity.See Edelman [41] for the theorem stating that σχ is indeed a closure operator.

Now for a transitive directed graph G = (V, A), a set C of arcs is a tran-sitive convex set (also known as t-convex set) if for every path P in C witha chord a ∈ A then a is in C. Given a transitive directed graph G = (V, A)

24 Chapter 3. An abstract notion of convexity

with C the set of t-convex sets of G the pair (V, C) is then called transitiv-ity convex geometry. Figure 3.12 gives the relation of inclusion between thedifferent families defined above.

Transitivity convex geometry

Affine convex geometry in R2

Lower affine convex geometry in R2 AOT convex geometry

FIGURE 3.12: The inclusion relation for different classes ofaffine convex geometries.

3.2.4 Search antimatroids in (directed) graphs

Given a directed graph G = (V, A) with a vertex r in V, we define an antima-troid on V \ rwhose feasible sets are F ⊆ V \ r such that in the subgraphinduced by F ∪ r every vertex of F can be reached from r by a directedpath. The pair (V \ r,F ) with F the set of feasible sets defined above isthe point-search antimatroid. Alternatively, we can define an antimatroidon A whose feasible sets are arc-sets of connected subgraphs containing r,the result is called line-search antimatroid. Those construction can be madeon undirected graphs, we obtain respectively the undirected point-searchantimatroid and the undirected line-search antimatroid.

It is possible to use both the vertices and edges as ground set, given agraph G = (V, E). We can define an antimatroid on V ∪ E where the collec-tion of feasible sets F is the collection of all F in 2V∪E such that if the edge(u, v) is in F, then either u or v is also in F. Then (V ∪ E,F ) is a point-linesearch antimatroid. Figure 3.13 gives us the relation of inclusion betweenthe different families defined above.

Undirected line-search antimatroid

Undirected point-search antimatroid

Point-search antimatroid

Line-search antimatroid

FIGURE 3.13: The inclusion relation for different classes ofsearch antimatroids in graphs.

3.2. Examples of convex geometries 25

3.2.5 Miscellaneous

Given a graph G = (V, E), Ardila and Maneva [8] mention a specific shellingprocess on V by deleting vertices of degree at most k. This shelling gives riseto an antimatroid on V if every induced subgraph of G contains at least onevertex with degree at most k.

In a graph, three vertices of a graph form an asteroidal triple if every twoof them are connected by a path avoiding the neighborhood of the third. Agraph is AT-free if it does not contain any asteroidal triple. Given an AT-freegraph G = (V, E), we define the interval I(u, v) between two vertices u and vas the set of vertices z in V for which there is a chordless u, z-path that avoidsN(v) ∪ v and a chordless v, z-path that avoid N(u) ∪ u. Chang et al. [27]use this notion to build a convex geometry on G. We define the set of convexC as follows, C ⊆ V is in C if and only if for every pair of vertices in C theinterval between those two vertices is also in C. The pair (V, C) is then calledan AT-free convex geometry.

Let V = 1, 2, . . . , n and 0 6 a, b 6 n integers. Define F ⊆ V2 recur-sively by ∅ ∈ F and if F ∈ F , j ∈ E \ F then F ∪ j is in F if and only ifthere are less than a numbers smaller than j not in F or less than b numberslarger than j not in F. The pair (V,F ) is an (a, b)-path shelling antimatroid.See Goecke et al. [54] for details. Poset antimatroids and double poset anti-matroids are special cases of (a, b)-path shelling antimatroids.

Let G = (V, E) be an undirected graph and let r /∈ V be an extra node.We call (V ∪r, C) an uncover convex geometry with C the union of the twosets: C ∪ r : C ∈ 2V and the set of vertex-sets that do not cover all edgesof G. Note that the circuits of this convex geometry are of the form (u, v, r)where u and v are adjacent.

Given a bipartite graph G = (V, E) with V = U1 ∪ U2 the vertex bi-partition and a two-coloring (red and blue) of the edges, we call a vertex vuniversal extreme in G if v is not incident to any red edge. Now we definea shelling process on U1 by iteratively deleting a universal extreme vertex(from U1) and its neighbors (in U2). This process gives an antimatroid on asubset of U1. This subset is not always equal to U1, in particular, if G has noblue edge, the antimatroid obtained is (∅, ∅).

Surprisingly, every antimatroid can be represented in this way. To seethis let (V,F ) be any antimatroid and denote by K the collection of circuitsof (V,F ). We build a bipartite graph on V ∪K. We connect v in V to a circuit

26 Chapter 3. An abstract notion of convexity

in K by a red edge if v is the root of this circuit and by a blue edge if v is anon-root element of this circuit. This defines a two-colored bipartite graphG = (V ∪ K, E) and v in V is universal extreme in this graph if and only ifv is not the root of any circuit of (V,F ). From this we see that the shellingprocess on G defined by iteratively deleting a universal extreme vertex givesthe antimatroid (V,F ). An example is given by Figure 3.14.

1, 2, 3, 4

2, 3, 4 1, 3, 4 1, 2, 4

2, 4 1, 4 1, 2

4 2 1

∅

F1

2

3

4

(1, 4, 3)

(2, 4, 3)

(1, 2, 4, 3)

FIGURE 3.14: A representation of an antimatroid by a two-colored bipartite graph.

27

Chapter 4

The maximum-weight convex setproblem

“Meticulous, yes. Methodical. Educated. They were these things. Nothingextreme. Like anyone, they varied. There were days of mistakes and laziness andinfighting. And there were days, good days, when by anyone’s judgment, theywould have to be considered clever. No one would say that what they were doingwas complicated. They took from their surroundings what was needed, andmade of it something more.”

— Aaron in Primer

In this chapter, we look at a fundamental optimization problem over con-vex geometries. The overall goal is to provide a polynomial-time algorithmfor finding a maximum-weight convex set in a convex geometry (providedwith a weight function) coming from a specific family. After having properlydefined the problem and given some context, we will study the special caseof monophonically convex geometries built on split graphs. The results inthis chapter were obtained in collaboration with Cardinal and Doignon [25].

4.1 A classic optimization problem

In many practical optimization problems, feasible solutions consist of one ormore sets that are required to satisfy some kind of convexity constraint. Theycan take the form of geometrically convex sets, such as in spatial planningproblems, e.g. Williams [122], electoral district design, e.g. Douglas et al. [78],or underground mine design, e.g. Parkinson [100]. Alternatively, convexitycan be defined in a combinatorial fashion.

28 Chapter 4. The maximum-weight convex set problem

4.1.1 Problem definition

Many classical problems in combinatorial optimization have the followingform. For a set system (V,X ) and for a function w : V → R, find a set X ofX maximizing the value of

w(X) = ∑x∈X

w(x).

For instance, the problem is known to be efficiently solvable for the inde-pendent sets of matroids, see Oxley [99], using the greedy algorithm. Sinceconvex geometries capture a combinatorial abstraction of convexity in thesame way as matroids capture linear dependence, we investigate the opti-mization of linear objective functions for convex geometries. It is not knownwhether a general efficient algorithm exists in the case of convex geometries.Of course, the hardness of finding a maximum-weight feasible set dependson the way we encode the convex geometries, see for instance Eppstein [45]and Enright [43] for more information. If one is given all the convex sets anda real weight for each element, it is trivial to find a maximum-weight convexset in time polynomial in |C| where C is the set of convex sets.

We investigate what happens if we choose a more compact way to encodethe information. We use now the copoint poset to describe a convex geom-etry. However, optimization on convex geometries given in this compactway is hard as Theorem 4.1.2 shows. Formally, we focus on the followingmaximum-weight convex set problem.

Problem 1. Given a convex geometry (V, C) encoded by its copoint posetand a weight function w : V → R, find a set C in C that maximizes the valueof w(C).

Note that, because the duality between convex and feasible sets and thefact that there are no sign restriction on the weights, an algorithm that solvesProblem 1 also solves directly the following maximum-weight feasible setproblem.

Problem 2. Given an antimatroid (V,F ) encoded by its base poset and aweight function w : V → R, find a set F in F that maximizes the value ofw(F).

4.1. A classic optimization problem 29

So we can now switch to the antimatroid point of view, thus making theexposition consistent with the historical development of the complexity re-sult below.

4.1.2 Computational hardness

We show here that solving Problem 2 (and also Problem 1) is hard. We firstrecall the following theorem due to Hastad [63], initially stated in terms of amaximum clique.

Theorem 4.1.1. There can be no polynomial time algorithm that approximates theproblem of finding the maximum size of an independent set in a graph on n verticesto within a factor better than O(n1−ε), for any ε > 0, unless P = NP .

For antimatroids given in the form of a base poset, we describe a NP-completeness reduction from the maximum independent set problem to themaximum-weight feasible set. The reduction is an adaptation of a result dueto Eppstein [46].

Theorem 4.1.2. The problem of finding a maximum-weight feasible set in an an-timatroid encoded in the form of its base poset is not approximable in polynomialtime within a factor better than O(N

12−ε) for any ε > 0, where N is the number of

elements in the base poset, unless P = NP .

Proof. Given any graph G = (V, E) on which we want to find an indepen-dent set of maximum size, we use G to build a point-line search antimatroid(A,F ) by letting A = V ∪ E and defining a feasible set (an element of F ) asany subset F of A such that v1, v2 ∈ E ∩ F then v1 ∈ F or v2 ∈ F. Remarkthat (A,F ) is indeed an antimatroid because it satisfies both the accessibilityproperty and the union stability, and A ∈ F .

The base poset of this antimatroid is composed of sets v for each vertexin V and sets v, e for each edge e ∈ E such that v ∈ e. Let d(v) denote thedegree of the vertex v and δ = 0.1. We define a weight function: w : A → R

by setting

w(x) =

−d(x) + δ if x ∈ V

1 if x ∈ E.

We first show that if F is a given feasible set with weight w(F), then we canconstruct an independent set of G of size at least w(F)δ−1 in polynomial time.To that end, we define a feasible set F′ ⊆ F as follows. If V ∩ F corresponds

30 Chapter 4. The maximum-weight convex set problem

to an independent set of vertices in the graph G, then F′ = F. If it is not thecase, we select a pair u, v ⊆ F such that e = u, v ∈ E, and remove oneelement of e from F. If u was removed, we also remove from F any elementu, a ∈ F ∩ E, with a /∈ F (to maintain the feasibility of the set). We repeatthis operation until the remaining vertices in the set F′ form an independentset in the graph. The remaining elements then form the set F′. It is easy tocheck that F′ is always feasible. By the definition of the function w, we havethe following inequalities,

w(F) 6 w(F′) 6 ∑v∈V∩F′

(−d(v) + δ) + ∑e∈E∩F′

1

6 ∑v∈V∩F′

(−d(v) + δ) + ∑v∈V∩F′

d(v) = δ|V ∩ F′|.

So we have an independent set V ∩ F′ that we can construct in polynomialtime with size at least w(F)δ−1.

Now, let N be the number of bases of (A,F ), and suppose we have afunction f and a f (N)-approximation algorithm to find a maximum-weightfeasible set, i.e. we have an algorithm that returns a feasible set with weightat least f (N)−1 times the weight of a maximum weight feasible set. Assumethat f (N) 6 O(N

12−ε) for some 0 < ε < 1. We know that N = |V|+ 2|E|, so

f (N) 6 O((|V|+ 2|E|) 12−ε) 6 O((|V|+ |V|2) 1

2−ε) 6 O(|V|1−ε′),

for an ε′ ∈]0, 1[. So we have

1f (N)

>1

O(n1−ε′),

and we obtain a feasible set with weight at least 1O(n1−ε′ )

times the weight w∗

of a maximum-weight feasible set. By the previous paragraph, we build anindependent set with size at least (w∗/O(n1−ε′))δ−1, so at least 1/O(n1−ε′)

the size of a maximum independent set, and this contradicts Theorem 4.1.1.So f (N) 6 O(N

12−ε′) is impossible unless P = NP .

Moreover, the above theorem remains true also for a subclass of antima-troids: the family of point-line search antimatroids.

4.2. Special cases solvable in polynomial time 31

4.1.3 Note on polyhedral results

A problem closely related to the optimization problem is the linear descrip-tion of the polytope defined by the convex sets. Korte and Lovasz [82] offerresults on the topic. We state the definitions below. Let (V, C) be a convexgeometry, the convex set polytope of (V, C)is defined by

conv(C) = conv(χC : C ∈ C),

where χC ∈ RV is the characteristic vector of C in C, i.e. for i in V, we have(χC)i = 1 if i ∈ C and (χC)i = 0 otherwise.

The study of convex set polytopes is not the topic of this work, but as wesee in the next section, knowing the linear description of conv(C) sometimeshelps in building an algorithm finding the maximum-weight convex set ofa convex geometry (V, C). Obviously, the feasible set polytope is definedusing the same technique.

4.2 Special cases solvable in polynomial time

We will now review optimization problems restricted to some specific convexgeometries, for which there is already a polynomial-time algorithm that findsa maximum-weight convex set.

4.2.1 Result for poset convex geometries

The problem of finding a maximum-weight convex set in a poset convex ge-ometry was first solved efficiently by Picard [104] using a minimum cut al-gorithm. For a poset convex geometry (V, C) built on the poset (V,6P), thepolytope conv(C) is given by

x ∈ RV : 0 6 xv 6 1 and xv 6 xu, ∀u, v ∈ V such that v 6P u.

The proof can be found in Picard [104] and Stanley [117]. With this descrip-tion, we have the following result from Picard [104] where TMCut(n, m) de-notes the time complexity for solving the minimum s, t-cut, in a s, t-graphwith n vertices and m arcs.

32 Chapter 4. The maximum-weight convex set problem

Theorem 4.2.1. For a poset P = (V,6P), and a weight function w : V → R,finding an ideal of P that minimizes w can be done in O(TMCut(|V|, m)) time,where m is the number of cover relations in P.

The minimum s, t-cut can be obtained in polynomial time, for instancesee Orlin [98] for a proof that TMCut(n, m) 6 O(nm). Note that switchingfrom a poset representation to a copoint poset can be done in polynomialtime. Below we give an idea of Picard’s reduction from the maximum flowproblem. First, we define an s, t-graph G that has a vertex xv for each elementv in V, a source vertex s and a sink vertex t. We now assign a weight on anyarc of G as follows. If w(v) > 0, then there is an arc from s to xv with capacityw(v). If w(v) < 0, then there is an arc from xv to t with capacity −w(v). LetV+ be the set of vertices with positive weights, and V− the set of nodes withstrictly negative weights. For each cover relation u 6P v with u, v in V wecreate an arc (u, v) with infinite capacity. This is illustrated by Figure 4.1. Forany s, t-cut s ∪ S, S ∪ t of the graph, we see that S is an ideal of theposet if there is no infinite capacity arc from S to S.

a −1 b −2

c −4 d 5

e 1

s

t

a b c d e

∞

∞

∞ ∞∞

1

5

1 2 4

FIGURE 4.1: Illustration of the Picard’s reduction.

If we call WS the sum of the capacity of arcs that start in S and end in S,we have

minS⊆V

WS = minS⊆V

∑v∈S∩V+

w(v) + ∑v∈S∩V−

−w(v).

= minS⊆V

∑v∈S∩V+

w(v)−(

∑v∈V−

w(v)− ∑v∈S∩V−

−w(v)

)= min

S⊆V∑

v∈Sw(v)− w(V−).

4.2. Special cases solvable in polynomial time 33

The last term is constant, so we find an ideal of minimum weight by usingan efficient minimum cut algorithm, for instance Orlin [98]. As the weightsare real, switching from a minimization problem to a maximization problemis direct.

4.2.2 Result for double poset convex geometries

Groflin [60] gives an algorithm to solve the maximum-weight convex setproblem for double poset convex geometries. The algorithm relies on the lin-ear description of the convex set polytope for those convex geometries. Thispolytope is uniquely described by the box inequalities and some “alternativevector” associated to chains of the poset.

An elegant reduction from the maximum-weight ideals problem to themaximum-weight convex set in double poset convex geometry is presentedby Queyranne and Wolsey [107]. We give the main idea of the reduction aftersome definitions. For a poset P = (V,6P) with a weight function w we definea new poset Q = (U,6Q) such that U = V′ ∪V′′ in which each element v inV has two copies v′ ∈ V′ and v′′ ∈ V′′. For u1, u2 in U the relation u1 6Q u2

holds if u1 6P u2 and one of the following is true, u1, u2 ∈ V′ or u1, u2 ∈ V′′

or u1 ∈ V′, u2 ∈ V′′. Roughly, the Hasse diagram of Q consists of twocopies of the Hasse diagram of P, connected by the relations v′ 6Q v′′ for allv in V, where v′ and v′′ are the copies of V in V′ and V′′ respectively. Theweights are defined in Q by the weight function wQ with wQ(v′) = −w(v)and wQ(v′′) = w(v). The following proposition gives us the reduction.

Proposition 4.2.1. For a poset P = (V,6P), and a weight function w : V → R, asubset T ⊆ U is a maximum-weight ideal in the associated poset Q = (V′ ∪V′′,6Q

) for the weight wQ, if and only if the set S = v ∈ V : v′ /∈ T and v′′ ∈ T is amaximum-weight convex set for the original weight function w of the poset convexgeometry built on P.

For a poset P = (V,6P) with n elements and m cover relations, the re-duction implies the creation of a poset Q with 2n elements and n + 2m coverrelations. Hence the following theorem, using the result from the Subsec-tion 4.2.1.

Theorem 4.2.2. For a poset P = (V,6), and a weight function w : V → R,finding a convex set in the double poset convex geometry built on P that maximizes

34 Chapter 4. The maximum-weight convex set problem

w can be done in O(TMCut(|V|, |V|+ m)) time, where m is the number of coverrelations in P.

Again in this case, switching from a poset representation to a copointposet can be done in polynomial time.

4.2.3 Result for tree convex geometries on vertices

Given a tree T = (V, E) finding an optimal subtree can be done in polyno-mial time as shows by Maffioli [88]. We describe the dynamic programmingtechnique that leads to the theorem below. The description uses the nota-tions of Magnanti and Wolsey [89]. Note that a connected subgraph in a treeis called a subtree.

Theorem 4.2.3. For a tree T = (V, E), and a weight function w : V → R, findinga subtree T that maximizes w can be done in O(|V|2) time.

We will consider here a tree T = (V, E) with a root r in V and focus offinding a convex set containing r that maximizes w, or returning the emptyset if every convex set containing r has negative weight. Let T(v) be the sub-tree of T rooted at v containing all nodes u for which the path from r to ucontains v. Let p(v), the predecessor of v, be the first node u (not equal tov) on the unique path in T connecting v and r. Let S(v) be the immediatesuccessors of node v, that is, all nodes u such that p(u) = v. Now let H(v)denote the optimal solution value of the rooted subtree problem defined onthe tree T(v). If v is a leaf of T, then H(v) = max0, w(v). The dynamic pro-gramming algorithm moves up the tree from the leaves to the root. Supposethat we have computed H(u) for all successors of node v, then we determineH(v) using the recursion

H(v) = max

0, w(v) + ∑u∈S(v)

H(u)

.

An optimal solution is then found by working backward from the root,eliminating every subtree T(v) encountered with H(v) = 0. Running thisprocedure on every vertex of the tree gives the result on tree convex geom-etry on vertices. In this case, starting from a tree, we can obtain a copointrepresentation for the convex geometry in in polynomial time.

4.2. Special cases solvable in polynomial time 35

4.2.4 Result for tree convex geometries on edges

Given a tree T = (V, E), Groflin and Liebling [61] give a minimal linear de-scription of the convex set polytope based on the notion of alternating vectorfor a tree. From the description, the authors build a primal-dual algorithmthat leads to the following theorem for tree convex geometry on edges.

Theorem 4.2.4. For a tree T = (V, E), and a weight function w : E → R, findinga convex set in the tree convex geometry on edges built on T that maximizes w canbe done in O(|V|2) time.

Again, starting from a tree we can obtain a copoint representation for theconvex geometry in in polynomial time.

4.2.5 Result for affine convex geometries in the plane

For a set P of weighted points in the plane, Eppstein et al. [47] found a poly-nomial time algorithm to find a maximum-weight convex set in the convexgeometry induced by P. Indeed, the authors first describe the following re-sult. Note that the convex geometries considered here are encoded with co-ordinates of points in R2, not by copoint posets.

Proposition 4.2.2. We can preprocess a point set P in the plane in O(|P|2) timeand space, such that afterwards, for each triangle in P, the number of points (and thesum of their weights) in any (query) triangle can be determined in constant time.

In this context, the authors use the following definition. A weight func-tion w of a point set P is decomposable if and only if for any polygon Pol =〈p1, . . . , pm〉 with p1, . . . , pm in P and any index 2 < i < m, we have

w(Pol) = f (w(〈p1, . . . , pi〉), w(〈p1, pi, pi+1, . . . , pm〉), p1, pi),

where the function f takes constant time to compute. In other words,when w is decomposable we can cut the polygon Pol in two along the linespanned by p1 and pi to obtain the weight of Pol from the weights of thetwo smaller polygons and some information on the cut segment. The weightfunction is called monotone decomposable if it is monotone in its first argu-ment. The next theorem is one of the main results of Eppstein et al. [47]

Theorem 4.2.5. Let w be a monotone decomposable weight function of a point setP with |P| = n. The convex hull of k points in P that minimizes or maximizes w

36 Chapter 4. The maximum-weight convex set problem

can be computed in O(kn3 + G(n)) time, where G(n) is the total time required tocompute w for the O(n3) possible triangles in the set.

This result can be adapted to the maximum-weight convex set problemfor affine convex geometries in the plane as shown by Bautista-Santiago etal. [12].

Theorem 4.2.6. For a point set P, and a weight function w : P → R, finding aconvex set that maximizes w can be done in O(|P|3).

More details and context for this problem can be found in Queyranne andWolsey [108].

4.3 The case of split graphs

We consider here a special case of chordal graph antimatroids: the splitgraph shelling antimatroids, i.e. chordal graph antimatroids built on a splitgraph. We show that the feasible sets of such an antimatroid relate to someposet shelling antimatroids constructed from the graph. We discuss a fewapplications, obtaining in particular a simple polynomial-time algorithm tofind a maximum-weight feasible set. We also provide a simple description ofthe circuits and the free sets. This special case will be useful to better under-stand the general case of chordal graph shelling antimatroids.

4.3.1 Characterization of the feasible sets

We recall from Chapter 3 that a split graph is a graph whose set of verticescan be partitioned into a clique and an independent set (the empty set is bothindependent and a clique). Here we assume that for every split graph, thepartition is given and we will denote by K and I the clique and the indepen-dent set, respectively. Note that split graphs are chordal graphs, and they arethe only chordal graphs to be co-chordal (i.e. the complement of the graph isalso chordal). Here is a useful characterization of the feasible sets in a splitgraph shelling antimatroid. Example 1 provides an illustration.

Proposition 4.3.1. Let G = (K ∪ I, E) be a split graph and (V,F ) be the splitgraph vertex shelling antimatroid defined on G. Then a subset F of vertices is feasiblefor the antimatroid if and only if N(F) induces a clique.

4.3. The case of split graphs 37

Proof. For the necessary condition (Fig. 4.2), suppose we have a simplicialshelling O = ( f1, . . . , f|F|) of a feasible set F such that N(F) does not induce aclique in G. Then, for some vertices v1 and v2 in N(F) we have v1, v2 /∈ E.Hence v1, v2 6⊆ K, since K is a clique. Assume without loss of generalitythat v1 ∈ I. Let f j be the first element in O such that f j, v1 ∈ E. As v1 ∈ I, bydefinition of a split graph, f j ∈ K and f j is adjacent to all other vertices of K.Then f j is not adjacent to v2 because f j must be simplicial in G \ f1, . . . , f j−1,so v2 must be in I. Now let ft be the first element of O such that ft, v2 ∈ E(notice j 6= t). Since v2 ∈ I, by a completely symmetric argument, we haveft ∈ K and ft, v1 /∈ E. Now a contradiction follows because, if j > t, thevertex ft is not simplicial in G \ f1, . . . , ft−1, and if t > j the vertex f j is notsimplicial in G \ f1, . . . , f j−1.

Ff j ∈ K ft ∈ K

v1 ∈ I v2 ∈ I

FIGURE 4.2: Illustration of the proof of necessary condition forProposition 4.3.1.

Reciprocally, suppose that we have a set of vertices F such that N(F) in-duces a clique (Fig. 4.3). We will build a simplicial shelling O on F with thehelp of the following three-set partition of F:

V1 = F ∩ I,

V2 = (F ∩ K) \ N(I \ F),

V3 = (F ∩ K) ∩ N(I \ F).

We arbitrarily order the elements in each of the sets V1, V2, V3 and concatenatethe orderings in this order to obtain the sequence O. By the definition of asplit graph, it is obvious that the elements of O in V1 ∪V2 fulfill the conditionof a simplicial shelling. If V3 = ∅, we are done. Otherwise, N(V3) \ F isa clique and so it has exactly one element i in I, because N(F) induces aclique and thus all elements of V3 are adjacent to this single element of I \

38 Chapter 4. The maximum-weight convex set problem

F. Therefore the elements of O in V3 fulfill the conditions of the simplicialshelling.

K

Ii

F

V1

V2 V3

FIGURE 4.3: Illustration of the proof of the sufficient conditionfor Proposition 4.3.1.

Example 1. Figure 4.4 below shows two split graphs on which we build asplit graph shelling antimatroid. The set F on the left (Fig. 4.4(a)) is a feasibleset and we see that N(F) defines a clique. On the right (Fig. 4.4(b)), we havea clique C and a possible simplicial shelling is proposed for a set of verticessuch that its neighborhood is C.

K

IF

N(F)

(a) F ∈ F ⇒ N(F) induces aclique.

K

I

C

4

1 2

3

5

(b) F ∈ F ⇐ N(F) inducesa clique.

FIGURE 4.4: Examples for Proposition 4.3.1.

Proposition 4.3.2. Let (V, E) be a graph and F be a subset of V. If N(F) is a clique,then V \ F is m-convex.

Proof. Assume N(F) is a clique. Proceeding by contradiction, we take twovertices v, w in V \ F for which there exists a chordless path (u0, u1, . . . , uk)

4.3. The case of split graphs 39

with v = u0 and uk = w having at least one vertex in F. Now we select iminimal and j maximal in 1, . . . , k − 1 such that ui and uj are in F. Thennecessarily ui−1 and uj+1 are adjacent, so the path has a chord, contradiction.

The converse of the implication in Proposition 4.3.2 does not hold evenif the graph is connected. Even more: V \ F being m-convex does not implythat the graph N induced on N(F) is a parallel sum of cliques (in other words,that N is the complement of a multipartite graph). Figure 4.5 below shows acounter-example based on a 2-connected, chordal graph.

F

FIGURE 4.5: Counter-example for the converse implication inProposition 4.3.2.

The converse of the implication in Proposition 4.3.2 holds if (V, E) is asplit graph (this follows from Proposition 4.3.1 and Section 3 of Farber andJamison [50]).

Corollary 4.3.1. Let G = (K ∪ I, E) be a split graph and (V,F ) be the split graphvertex shelling antimatroid defined on G. For all feasible sets F, there is at most onei ∈ I \ F such that there is k ∈ K ∩ F with k, i in E.

Proof. This comes directly from Proposition 4.3.1 and the fact that the set I isan independent set.

Corollary 4.3.2. Let G = (K ∪ I, E) be a split graph and (V,F ) be the split graphvertex shelling antimatroid defined on G. Let u and v be distinct elements in V.Then V \ u, v ∈ F if and only if u, v is in E, or at least one of the two verticesis isolated in G.

Proof. This comes directly from Proposition 4.3.1.

Corollary 4.3.2 helps us to rebuild the original split graph for a given splitgraph shelling antimatroid, as shown in the following proposition.

40 Chapter 4. The maximum-weight convex set problem

Proposition 4.3.3. Let (V,F ) be a split graph shelling antimatroid with F 6= 2V ,then there is a unique split graph G such that (V,F ) is the split graph shellingantimatroid defined on G.

Proof. Suppose we have obtained a graph G such that (V,F ) is the splitgraph shelling antimatroid defined on it. Because F 6= 2V and Proposi-tion 4.3.1, the graph G must have a non-empty subset S of vertices such thatthere exist a, b in N(S) with a, b /∈ E (thus V \ a, b /∈ F ).

If we take an element v in V such that V \ u, v ∈ F for all u in V \ v(so v /∈ a, b), then Corollary 4.3.2 leaves two options. Either the vertexv is isolated in G or v forms an edge with every non-isolated vertex in G.Moreover, if this element v is such that v ∈ F , then the existence of thesubset S in the graph and Proposition 4.3.1 imply that v must be an isolatedvertex in the graph.

We now build the graph G = (V, E) as follows. First, we identify theisolated vertices as the vertices i satisfying V \ i, u ∈ F for all u in V \ iand i ∈ F . Next, among all pairs of non-isolated vertices v1, v2, the onesthat give an edge in G satisfy V \ v1, v2 ∈ F . We know that there is noother edge by Corollary 4.3.2.

Remark that for the split graph shelling antimatroid (V, 2V), there existseveral split graphs such that (V, 2V) is the split graph shelling antimatroiddefined on it. For instance the complete graph on V, or the graph (V,∅).

Testing whether a given antimatroid (V,F ) is a split graph shelling anti-matroid can be done using arguments in the proof of Proposition 4.3.3: First,build a graph G = (V, E) with v1, v2 ∈ E exactly if V \ v1, v2 ∈ F andV \ v1, u1 /∈ F for some u1 in V \ v1 and V \ v2, u2 /∈ F for some u2 inV \ v2. Next check that G is split and F consists of exactly the feasible setsof G.

We now distinguish two classes of feasible sets for the split graph shellingantimatroids. A feasible set F is an i-feasible set if there is a vertex i in N(F)∩I (by Corollary 4.3.1, such an i is unique). On the other hand, a feasible setF is a ∗-feasible set when N(F) ⊆ K. Figure 4.6 below illustrates the twoclasses of feasible sets.

A feasible set of a split graph shelling antimatroid belongs either to thefamily F ∗ of ∗-feasible sets, or to one family F i of i-feasible sets as shown inthe following corollary. The proof is straightforward.

4.3. The case of split graphs 41

K

I

F2

i

(a) F2 is an i-feasible set.

K

I

F1

(b) F1 is a ∗-feasible set.

FIGURE 4.6: Examples of the two classes of feasible sets.

Corollary 4.3.3. Let G = (K ∪ I, E) be a split graph and (V,F ) be the split graphvertex shelling antimatroid defined on G. If I = i1, . . . , i|I|, then F decomposesinto

F ∗ ∪ F i1 ∪ · · · ∪ F |I|.

4.3.2 Connection between split graph shellings and posets

For investigating a split graph (K ∪ I, E), we will make use of two functionsfrom I to 2K∪I , the forced set function and the unforced set function, respec-tively:

fs(i) =k ∈ K : k, i /∈ E ∪ i′ ∈ I : N(i′) 6⊆ N(i),uf(i) = fs(i) \ i = k ∈ K : k, i ∈ E ∪ i′ ∈ I : N(i′) ⊆ N(i) \ i.

As shown in the next lemma, the forced set function evaluated at i gives usthe vertices which belong in any i-feasible set. The unforced set functionevaluated at i just gives the complement of fs(i), minus i. So for all i in I thevertex set of the graph is equal to i ∪ fs(i) ∪ uf(i). Those two definitionsare illustrated in Figure 4.7.

Lemma 4.3.1. Let G = (K ∪ I, E) be a split graph and (V,F ) be the split graphvertex shelling antimatroid defined on G. Let i be in I, then for any i-feasible set Fin F we have fs(i) ⊆ F.

Proof. If F is an i-feasible set, we have i ∈ I ∩ F and there is k in K ∩ F withk, i ∈ E. If a vertex v in fs(i) is not in F then we have two possibilities.Either v ∈ K and so i, v /∈ E (by definition of fs(i)), but that contradicts

42 Chapter 4. The maximum-weight convex set problem

K

I

uf(i2) fs(i2)

k1 k2

k3

i1i2i3

FIGURE 4.7: Illustration of the forced set and unforced set func-tions.

Proposition 4.3.1 because i, v ⊆ N(F), or v ∈ I and there is k′ ∈ K such thatk′, i /∈ E but k′, v ∈ E (by definition of fs(i)). We know that k, k′ ∈ E bydefinition of K, but that also contradicts Proposition 4.3.1 because if k′ /∈ F,then i, k′ ⊆ N(F), and if k′ ∈ F then i, v ⊆ N(F) with i, v /∈ E bydefinition of I.

Lemma 4.3.2. Let G = (K ∪ I, E) be a split graph and (V,F ) be the split graphvertex shelling antimatroid defined on G. Let i be in I, then fs(i) ∈ F ∗.

Proof. Directly from the definition of fs(i), we have that N(fs(i)) ⊆ K. Sofs(i) ∈ F ∗.

We will now establish the link between split graph shelling antimatroidsand poset antimatroids. If we have a split graph G = (K ∪ I, E), we builda poset on K ∪ I with the binary relation ≺ defined by u ≺ v if and only ifu ∈ K, v ∈ I and u, v ∈ E in G. The resulting poset (K∪ I,≺) is of height atmost two (the number of elements in a chain is at most two). Next, we provethat all the structures (V,F ∗) and (uf(i), F \ fs(i) : F ∈ F i ∪∅) for i in Iare poset antimatroids.

Proposition 4.3.4. Let G = (K ∪ I, E) be a split graph and (V,F ) be the splitgraph vertex shelling antimatroid defined on G, then F ∗ = flt(K ∪ I,≺).

Proof. First we show that F ∗ ⊆ flt(K ∪ I,≺). Take F in F ∗, by definition of a∗-feasible set, if there is a k ∈ F ∩ K, then N(k) ∩ I ⊆ F. Then F is a filter of(K ∪ I,≺).

Next we show that F ∗ ⊇ flt(K ∪ I,≺). Suppose that F is a filter of (K ∪I,≺), then N(F) ⊆ K by the definition of ≺, and by Proposition 4.3.1 we

4.3. The case of split graphs 43

know that F is a feasible set because K is a clique, and also a ∗-feasible setbecause N(F) ∩ I = ∅.

In the following, we use (uf(i),≺) to denote the poset formed on uf(i)with the binary relation ≺ restricted to uf(i).

Proposition 4.3.5. Let G = (K ∪ I, E) be a split graph and (V,F ) be the splitgraph vertex shelling antimatroid defined on G, then for all i in I, F i = fs(i)∪H :H ∈ flt(uf(i),≺) , H ∩ K 6= ∅.

Proof. Let i be in I. We first show that F i ⊆ fs(i) ∪ H : H ∈ flt(uf(i),≺) , H ∩ K 6= ∅. Take a F in F i, then fs(i) ⊆ F by Lemma 4.3.1. We havedirectly from the definition of F i that (F \ fs(i)) ∩ K 6= ∅, note also thatF \ fs(i) = F ∩ uf(i). Now we just need to show that F ∩ uf(i) is a filter of(uf(i),≺). This is equivalent to showing that N(k) ∩ I ∩ uf(i) ⊆ F for all k inK ∩ (F ∩ uf(i)). So if we take a k in K ∩ (F ∩ uf(i)), then k, i ∈ E, but F is afeasible set, so we must have, by Corollary 4.3.1, N(k) ∩ I ∩ uf(i) ⊆ F.

Secondly, we show that F i ⊇ fs(i) ∪ H : H ∈ flt(uf(i),≺) , H ∩ K 6= ∅.Suppose that H is a filter of (uf(i),≺) such that H ∩K 6= ∅. We need to showthat H ∪ fs(i) is a feasible set. We use again Proposition 4.3.1 and check thatN(fs(i) ∪ H) induces a clique. We only need to observe that N(fs(i)) ⊆ N(i)by definition of the function fs, and this implies N(fs(i) ∪ H) ⊆ N(i) ∪ iwhich is a clique. Finally, by construction H ∩ N(i) 6= ∅ and i /∈ H, sofs(i) ∪ H is an i-feasible set and the proof is complete.

Note that, in the above proposition, for i in I the set fs(i) ∪ H : H ∈flt(uf(i),≺) without the condition H ∩ K 6= ∅ is included in F ∗ ∪ F i.

Corollary 4.3.4. Let G = (K ∪ I, E) be a split graph and (V,F ) be the split graphvertex shelling antimatroid defined on G, then (V,F ∗) and (uf(i), F \ fs(i) : F ∈F i ∪∅) for i in I are all poset antimatroids.

Proof. This follows directly from Propositions 4.3.4 and 4.3.5.

Proposition 4.3.5 shows us that an i-feasible set can be decomposed intofs(i) and a filter of (uf(i),≺). It is easy to see that this decomposition isunique.

The definitions of ∗-feasible and i-feasible sets lead to a better understand-ing of the poset produced by the feasible sets of a split graph shelling anti-matroid. Indeed, for a split graph shelling antimatroid (V,F ) built on a split

44 Chapter 4. The maximum-weight convex set problem

graph (K ∪ I, E), we decompose the structure of the poset formed by its fea-sible sets into a poset (F ∗,⊆) and |I| posets (F i,⊆), for i ∈ I, as illustratedby Figure 4.8 and detailed in Example 2.

∅(F ∗,⊆)

fs(i)

fs(i′)(F i,⊆)(F i′ ,⊆)

FIGURE 4.8: Schematic view of the poset produced by the fea-sible sets.

Example 2. Figure 4.9 shows a split graph (K ∪ I, E) and the poset formed bythe feasible sets of the split graph shelling antimatroid built on it. The dashedlink between a feasible set F and the union sign means that above this point,we look at i-feasible sets (for some i ∈ I) and the sets considered must betaken in union with fs(i).

4.3.3 The base poset

We recall from Chapter 3 that for an antimatroid, a base is a feasible set thatcannot be decomposed into the union of two other (non-empty) feasible sets.For a split graph shelling antimatroid (V,F ) built on a split graph (K ∪ I, E),the base poset is easy to obtain in terms of the following sets:

P1 = i : i ∈ I,P2 = k ∪ (N(k) ∩ I) : k ∈ K,P3 = fs(i) ∪ k ∪ (N(k) ∩ I \ i) : i ∈ I, k ∈ N(i),P = P1 ∪ P2 ∪ P3.

Proposition 4.3.6. Given a split graph shelling antimatroid (V,F ) built on a splitgraph (K∪ I, E) without any vertex i in I such that N(i) = K, its set of bases equalsP.

In the proposition above, the condition forbidding any i in I such thatN(i) = K is not very restrictive because if such an i exists, then we changethe partition K ∪ I into (K ∪ i) ∪ (I \ i).

4.3. The case of split graphs 45

K

I

12

3

4 5 6

∅

4 5 6

4, 5 5, 6 4, 6 3, 6

1, 4, 5 2, 5, 6 4, 5, 6 3, 5, 6 3, 4, 6

1, 4, 5, 6 2, 4, 5, 6 3, 4, 5, 6 2, 3, 5, 6

1, 2, 4, 5, 6 2, 3, 4, 5, 6 1, 3, 4, 5, 6

1, 2, 3, 4, 5, 6

(F ∗,⊆)

∪

2

2, 41, 4

1, 2, 4

(F 5,⊆)

∪

1

(F 4,⊆)∪

2 3

2, 3

(F 6,⊆)

FIGURE 4.9: A split graph and the feasible sets posets associ-eted with its shelling antimatroid.

Proof. Every set in P is feasible, because of Proposition 4.3.1. Next we showthat every feasible set in P cannot be decomposed into the union of twoproper feasible sets. This is trivial for the sets in P1. So suppose that F =

k ∪ (N(k) ∩ I) in P2 is the union of two proper feasible sets F1 and F2

with k ∈ F1. Then F1 is not feasible because of Proposition 4.3.1 and the as-sumptions which ensures that there is no i in I such that N(i) = K. Nowsuppose that F = fs(i) ∪ k ∪ (N(k) ∩ I \ i) in P3 is the union of twoproper feasible sets F1 and F2 with k ∈ F1. Then, using Proposition 4.3.1, F1 isnot feasible because i ∈ N(F1) but for all f in F \ k we have f , i /∈ E, soN(F1) do not induces a clique.

Second, we show that every F in F is the union of some sets in P. If F is a∗-feasible set, we use the sets from P1 and P2. If F is an i-feasible set, we usethe sets from P3 of the form fs(i) ∪ k ∪ (N(k) ∩ I \ i) with k ∈ N(i) andsome sets from P1. By Corollary 4.3.3 we are done.

Figure 4.10 illustrates Proposition 4.3.6. Directly from Proposition 4.3.6,we have the following corollary.

46 Chapter 4. The maximum-weight convex set problem

K

I

12

3

4 5 6

4 5 6

1, 4, 5 2, 5, 6 3, 6

1, 2, 4, 5 1, 3, 4, 5 2, 3, 6 1, 3, 4, 6

1, 2, 3, 5, 6

P1

P2

P3

FIGURE 4.10: A split graph and the base poset associated withits shelling antimatroid.

Corollary 4.3.5. Let (V,F ) be a split graph shelling antimatroid built on a splitgraph (K ∪ I, E) without any vertex i in I such that N(i) = K. The number of basesin (V,F ) is at most the number of vertices plus the number of edges from K to I.

4.3.4 Optimization results

We will now prove that for a weighted split graph shelling antimatroid, theproblem of finding a maximum-weight feasible set can be done in polyno-mial time in the size of the input even if the form of the input considered isa more compact representation than the base poset. We use the split graphitself to encode all the information about the feasible sets.

In the case of the poset antimatroids, the optimization problem (intro-duced in Subsection 4.2.1) is solved using the solution to the maximum clo-sure problem:

Problem 3. Given a poset (V,6) and a weight function w : V → R, find afilter F that maximizes

w(F) = ∑f∈F

w( f ).

We recall from Section 4.2 that Picard [104] designs a polynomial algo-rithm to solve Problem 3. The method runs in O(TMCut(n, m)) time, where

4.3. The case of split graphs 47

TMCut(n, m) denotes the time complexity for solving the minimum s, t-cut,in a s, t-graph with n vertices and m arcs. Moreover, we know from Orlin [98]that TMCut(n, m) 6 O(nm). From now, we denote by TMClo(n, m) the timecomplexity for solving the maximum closure problem for a poset with n el-ements and m cover relations. Taking advantage of this optimization result,we have the following theorem.

Theorem 4.3.1. Giving a split graph G (as a list of vertices and a list of edges),the problem of finding a maximum-weight feasible set in the split graph shellingantimatroid defined on G can be done in polynomial time.

Proof. We recall that for every split graph shelling antimatroid we introducea unique poset with relation ≺ (see just before Proposition 4.3.4) . The con-struction of this poset combined to Corollary 4.3.3 and Propositions 4.3.4and 4.3.5 allows us to decompose the problem of finding a maximum feasi-ble set in a split graph antimatroid into several maximum closure problems.Indeed, we first solve the maximum closure problem for (V,≺), yielding a∗-feasible set with maximum weight among all the ∗-feasible sets. Then foreach i in I, we solve the maximum closure problem for (uf(i),≺), yielding aset S such that S ∪ fs(i) is a feasible set (in F ∗ ∪ F i) that has greater or equalweight than each i-feasible set. The algorithm outputs the feasible set foundwith maximum weight.

So suppose that we have a procedure to find a filter in a poset (V,6) ofmaximum weight (given by a function w) called MaxClo(V,6, w). In a splitgraph (K ∪ I, E), we look at the element i in I that maximizes the weight offs(i) ∪MaxClo(uf(i),≺, w), we then compare the result with the weight ofMaxClo(K ∪ I,≺, w) and keep the maximum. The time complexity of thealgorithm is O(|I|TMClo(|I|+ |K|, |E|)). Note that if we use a procedure tofind a filter in a poset (V,6) of minimum weight (given by a function w),with very little modifications, our algorithm can be used to return a feasibleset of minimum weight.

4.3.5 Free sets and circuits characterization

Let G = (V, E) be a chordal graph. It is known that the rooted circuits ofits vertex shelling antimatroid admit the following simple description: a pair(C, r) is a rooted circuit if C consists of two distinct vertices u, v such that r is

48 Chapter 4. The maximum-weight convex set problem

an internal vertex on some chordless path joining u and v (this follows imme-diately from Corollary 3.4 in Farber and Jamison [50]). Moreover, the circuit(C, r) is critical if and only if the path has exactly three vertices. For the par-ticular case of split graphs we now provide more efficient characterizationsof (critical) circuits, and then of free sets.

Proposition 4.3.7. Let (V,F ) be the vertex shelling antimatroid of the split graph(K ∪ I, E). Set

C1 =(i, j, k) : k ∈ K, i, j ∈ N(k) ∩ I;C2 =(i, l, k) : k ∈ K, i ∈ N(k) ∩ I, l ∈ (N(k) ∩ K) \ N(i);C3 =(i, j, k) : k ∈ K, i ∈ N(k) ∩ I, j ∈ I \ N(k) and

∃m ∈ K with i, m /∈ E, j, m ∈ E.

Then the collection of rooted circuits of (V,F ) equals C1 ∪ C2 ∪ C3. Moreover, thecollection of critical rooted circuits equals C1 ∪ C2.

Proof. Notice that any chordless path in a split graph (K ∪ I, E) has at mostfour vertices. Moreover, if it has three vertices, the internal vertex is in K andat least one endvertex is in I. If it has four vertices, the internal vertices arein K and the endvertices in I. The result then follows from the characteriza-tion of the circuits of the shelling antimatroid of a chordal graph (which werecall just before the statement): the rooted circuits forming C1 and C2 comefrom paths with three vertices, those forming C3 come from paths with fourvertices.

Proposition 4.3.8. Let G = (K ∪ I, E) be a split graph with L and J (possiblyempty) subsets of respectively K and I. Then L ∪ J is free in the vertex shellingantimatroid of G if and only if either there is no edge between L and J, or there existssome vertex h in J such that L ⊆ N(h) and N(J \ h) ⊆ N(h) \ L.

Proof. Assuming first that X is a free set in (V,F ), we let L = X ∩ K and J =X ∩ I (we may have L and/or J empty). If no edge of V has an endvertex in Land the other one in J, then L∪ J is as in the first case of the statement. If thereis some edge l, h with l ∈ L and h ∈ J, we show that L and J are as in thesecond case of the statement. First, there holds L ⊆ N(h) because otherwisefor any vertex u in L \ N(h), we would find the circuit h, l, u in X (but afree set cannot contain any circuit). Second, we prove N(J \ h) ⊆ N(h) \ Lagain by contradiction. Thus assume some vertex v belongs to N(J \ h) \

4.3. The case of split graphs 49

(N(h) \ L). Then v is adjacent to some i in J \ h, and v belongs to eitherL or K \ N(h). In the first eventuality, X contains the circuit h, v, i. In thesecond eventuality, whether l, i ∈ E or l, i /∈ E, the circuit h, l, i is inX. In both eventualities we reach a contradiction. Thus X = L ∪ J is as in thesecond case of the statement.

Conversely, assume L and J are as in the statement and let us prove thatX = L∪ J contains no circuit, and so that X is free. If a rooted circuit (i, j, k)from C1 (as in Proposition 4.3.7) is in X, our assumption imposes i = h = j, acontradiction. If a rooted circuit (i, l, k) from C2 is in X, then our assump-tion implies first i = h because l ∈ N(i) ∩ L, and then k /∈ L in contradictionwith k ∈ X. Finally, if a rooted circuit (i, j, k) from C3 is in X with m as inC3, our assumption implies i = h, but then m ∈ N(k) \ N(h) is a contradic-tion with the assumption.

4.3.6 Beyond this special case

We studied the structure of split graph shelling antimatroids and describedan algorithm to solve the maximum-weight feasible set problem in polyno-mial time. The antimatroids considered form a very special class, but theidea of using the maximum closure problem for a poset can be generalized tosolve the maximum convex set problem for monophonically convex geome-try as we see in the next chapter.

51

Chapter 5

Finding a maximum-weight convexset in a chordal graph

“I was there to push people beyond what’s expected of them. I believe that’s anabsolute necessity.”

— Terence Fletcher in Whiplash

In this chapter, we consider the problem of finding a maximum-weightconvex set of a given vertex-weighted chordal graph. It generalizes previ-ously studied special cases in split graphs. It also happens to be closely re-lated to the closure problem in partially ordered sets and directed graphs.We give the first polynomial-time algorithm for the problem. This chapter isthe result of a collaboration with Cardinal and Doignon [24].

5.1 More on chordal graphs

Before developing the main result, we present some structural results onchordal graphs.

5.1.1 Definitions

In the following, monophonically convex sets of a chordal graph G will bereferred to as convex sets of G. From a path p = (v1, . . . , vn) we can extract achordless path by taking a shortest path between v1 and vn in the subgraphinduced by the vertices in p. A separator S of a graph G is a set of verticessuch that there exist two vertices u, v in V \ S connected by a path in thegraph but not in G − S. We say that S is a minimal separator if there is noseparator S′ of G such that S′ ⊂ S. For u, v in V, a subset S of V \ u, v is

52 Chapter 5. Finding a maximum-weight convex set in a chordal graph

a uv-separator if u and v are connected in G but not in G− S. The set S is aminimal vertex separator of G if S is a uv-separator for some u, v in V and Sdoes not strictly contain any uv-separator. Note that any minimal separatoris also a minimal vertex separator, but the converse does not hold in general.We denote by SG the set of all minimal vertex separators in G. Note that inchordal graphs, every minimal vertex separator is a clique. We also recall thatfor any chordal graph G = (V, E) we have |KG| 6 |V| and |SG| 6 |V| − 1,the proofs of those inequalities can be found in Fulkerson and Gross [52], andHo and Lee [64] respectively.

5.1.2 The clique-separator graph

Ibarra [69] introduces the clique-separator graph for chordal graphs. For achordal graph G, he defines a mixed graph where the nodes are the maximalcliques and minimal vertex separators of G. Moreover, the (directed) arcscapture the inclusion-covering relation between the minimal vertex separa-tors, while the (undirected) edges represent (minimal) inclusion from min-imal vertex separators to maximal cliques of G. More precisely, the clique-separator graph G of a chordal graph G has a set of clique nodes, one foreach maximal clique of G and a set of separator nodes one for each minimalvertex separator of G. The clique-separator graph has also a set A of edgesand arcs defined as follow. Each arc (S, S′) is from a separator node S to aseparator node S′ such that S ⊂ S′ and there is no separator node S′′ suchthat S ⊂ S′′ ⊂ S′. Each edge K, S is between a clique node K and a sep-arator node S such that S ⊂ K and there is no separator node S′ such thatS ⊂ S′ ⊂ K. Later in this work, we will denote by ArG the set of arcs in aclique-separator graph G. Figure 5.1 gives an example of a clique-separatorgraph of a chordal graph. Two of the mains results obtained by Ibarra are thefollowing theorems.

Theorem 5.1.1. For a chordal graph G = (V, E), constructing the clique-separatorgraph of G can be done in O(|V|3) time.

Theorem 5.1.2. Let G = (V, E) be a chordal graph with clique-separator graph Gand let S be a separator node of G. If G− S has connected components G1, . . . , Gt,then t > 1 and G − S′ : S′ ∈ SG, S′ ⊆ S has connected components G1, . . . ,G t

such that for every 1 6 i 6 t, the vertex set of Gi is the same as the vertex setrepresented by the nodes of G i − S.

5.2. Problems 53

G

K1

K2

K3

K4 K5

K7

K8 K9

K10 K11 K12

K6

G

K2 S1

K1

S2

K3

S4

K4 S5 S6 K6K5

S3 K7

S7

S8K8 K9

K11S9 S10K10 K12

FIGURE 5.1: A clique-separator graph G of a chordal graph G.

5.2 Problems

We formally introduce the optimization problems we will consider in the restof this chapter.

5.2.1 Main problem

Here is our main problem, the maximum-weight convex set problem in chor-dal graphs.

Problem 4. Given a chordal graph G and a weight function w : V → R, finda set C in CG that maximizes the value of w(C).

Our main result follows.

Theorem 5.2.1. The maximum-weight convex set problem in chordal graphs can besolved in polynomial time.

The well-known problem of finding a maximum-weight subtree (i.e. anacyclic connected subgraph) in a tree can be solved by selecting a vertex as“root”, finding a maximum-weight subtree that contains the root, and iterat-ing this procedure for all possible roots (see Magnanti and Wolsey [89]). Inorder to use a similar approach to solve Problem 4, we define a notion of root.It will be easier to work with chordal graphs which are connected. Note thatour results straightforwardly extend to the non-connected case.

54 Chapter 5. Finding a maximum-weight convex set in a chordal graph

5.2.2 Dummy vertices and sub-problems

In order to simplify some of the later statements and arguments, we wantto have in each maximal clique some vertex which is adjacent to no vertexoutside the clique and which has weight zero. To this aim, we add such avertex to any maximal clique (without changing the result of the optimizationproblems, see the end of the present subsection). Formally, let G = (V, E) bea vertex-weighted graph. For each maximal clique K of G, we add a newvertex dK to the graph and we make dK adjacent to exactly the vertices inK (unless K already has vertices of weight zero, not adjacent to any vertexoutside of K, then we select one of those that becomes dk). The weight of dK

is set to 0, while the other vertices keep their weight. The resulting vertex-weighted graph is called the extension G′ of G. Notice that the maximalcliques of G′ are all of the form K ∪ dK, where K is a maximal clique ofG; we call dK the dummy vertex of the maximal clique K ∪ dK. Given avertex-weighted chordal graph G = (V, E), its extension G′ = (V′, E′) is alsoa vertex-weighted chordal graph. Remark that G and G′ essentially have thesame clique-separator graph. When G = G′, we say that the vertex-weightedchordal graph G is extended.

G′

1

2

3 4 5

G

1

2

3 4 5

d2,3,4 d2,4,5

d1,2

FIGURE 5.2: A chordal graph and its extension.

For a set R of vertices of an extended vertex-weighted chordal graph G,we say that a convex set C of CG is R-rooted if R ⊆ C. If R is a singletonr we write r-rooted instead of r-rooted. This modification allows us todefine the following problem.

Problem 5. Given an extended chordal graph G with a weight function w :V → R and a maximal clique K of G, find a dK-rooted convex set C of G thatmaximizes the value of w(C).

We show below that, given any vertex-weighted chordal graph G, solvingProblem 5 for the extension G′ of G for all K in KG gives us a solution to

5.3. A special case solvable in polynomial time 55

Problem 4. The first lemma states the obvious link between the convex setsof G and G′.

Lemma 5.2.1. Let G = (V, E) be a chordal graph, C be a convex set of G and C′ bea convex set of G′ = (V′, E′), the extension of G. Then C is a convex set of G′ andC′ ∩V is a convex set of G.

Proof. First, C is a convex set of G′ because any chordless path in G′ betweentwo vertices of C is a chordless path in G. Second, C′ ∩ V is convex in Gbecause any chordless path in G between two vertices of C′ ∩V is a chordlesspath in G′.

The next lemma shows a stronger result than what we need for provingthe equivalence between Problem 4 and Problem 5, but it will be useful.

Lemma 5.2.2. Let G = (V, E) be a chordal graph with a convex set C in G, and G′

be the extension of G. Let KC be a maximal clique of the graph induced by C. Then,for every K′ in KG′ such that KC ⊆ K′, the set dK′ ∪ C is convex in G′.

Proof. For K′ in KG′ such that KC ⊆ K′, suppose that dK′ ∪ C is not convexin G′. So there is a chordless path (dK′ , f1, . . . , ft, c) in G′ with c in C butf1, . . . , ft not in C. Because f1 must be in K′, we know that for all v in KC wemust have v, c ∈ E (otherwise any chordless path in G we can extract from(v, f1, . . . , ft, c) contradicts the convexity of C). There results a contradictionwith the maximality of KC.

Lemmas 5.2.1 and 5.2.2 combined show that any algorithm solving Prob-lem 5 in polynomial time establishes Theorem 5.2.1. Indeed, we run the al-gorithm solving Problem 5 on every maximal clique and save a maximum-weight solution C∗ among all the outputs of the executions. Then we removethe dummy vertices from C∗ and we are done.

In what follows, the chordal graphs we consider are extended: we con-sider that every maximal clique K contains a fixed, dummy vertex dK.

5.3 A special case solvable in polynomial time

In this section, we solve Problem 5 for a family of special instances. We firstdefine a partial order relation on the vertices of a given chordal graph. Thenwe use this relation to reduce instances of Problem 5 in this family to theclosure problem in posets. The latter problem can be solved in polynomialtime using Picard’s algorithm [104].

56 Chapter 5. Finding a maximum-weight convex set in a chordal graph

5.3.1 The rooted poset

Let K be a maximal clique of a chordal graph G = (V, E). We define thebinary relation 6K on V as the set of pairs (u, v) ∈ V ×V such that there is achordless path (v, . . . , dK) that contains u. For the reduction we need to checkthat the relation is indeed a partial order.

Theorem 5.3.1. For G = (V, E) a chordal graph and K a maximal clique of G, thepair (V,6K) is a poset.

It can be shown that the order relation we just defined is a special case ofthe C-factor relation defined by Edelman and Jamison [42] (taking the convexset C equal to dK). The poset PK = (V,6K) will be referred to as the K-rooted poset of G. Figure 5.4 shows a chordal graph and the Hasse diagramfor (V,6K) with K = 1, 2, d1,2.

Theorem 5.3.1 directly follows from the two lemmas below, respectivelystating that the relation is antisymmetric and transitive, and whose proofsare illustrated in Figure 5.3. The reflexivity of the relation is obvious.

Lemma 5.3.1 (Antisymmetry). For G = (V, E) a chordal graph and K a maximalclique of G, the relation 6K is antisymmetric.

Proof. For a and b in V, we show that we cannot have a 6K b, b 6K aand a 6= b. Suppose a 6K b and b 6K a, so there are two chordless paths(u1, . . . , uj, . . . , un) and (v1, . . . , vl, . . . , vm) with u1 = vl = a, uj = v1 = b andun = vm = dK. If we take the path

(uj+1, . . . , un, vm−1, . . . , vl, u2, . . . , uj−1),

we can extract a chordless path p with starting vertex uj+1 and ending vertexuj−1. The path p has at least three vertices because uj+1, uj−1 is not in E.The vertex b is not in p because a 6= b, so we can add it to p. But then,a contradiction arises, because we obtain a chordless cycle with more thanthree vertices due to the fact that b only forms an edge with uj−1 and uj+1

among the considered vertices.

Lemma 5.3.2 (Transitivity). For G = (V, E) a chordal graph and K a maximalclique of G, the relation 6K is transitive.

Proof. For a, b and c three different vertices in V, suppose we have a 6K b andb 6K c. So we have (u1, . . . , uj, . . . , un) and (v1, . . . , vl, . . . , vm), two chordless

5.3. A special case solvable in polynomial time 57

paths with u1 = a, uj = v1 = b, vl = c and un = vm = dK. From the path(u1, . . . , uj, v2, . . . , vm) we extract a chordless path (u1, . . . , ux, vy, . . . , vm). Ify ∈ 1, . . . , l then c is in the chordless path and we have a 6K c. If y ∈l + 1, . . . , m, then from the following path:

(uj+1, . . . , un, vm−1, . . . , vy, ux, . . . , uj−1),

we can extract a chordless path p that avoids b, with starting vertex uj+1 andending vertex uj−1. A similar argument to the proof of the previous lemmashows a contradiction.

a,u1,vl b, uj, v1

dK

vm−1

uj−1 uj+1

a,u1 b, uj, v1

vm−1

dKuj−1 uj+1

ux

vy

c

FIGURE 5.3: Illustrations of the proofs of antisymmetry andtransitivity for the relation 6K.

G = (V, E)

1

23

45

6 7

8

d1,2

d3,4,5

d4,5,7

d2,3,4

d4,6,7

d6,8

(V,61,2,d1,2

)

d1,2

1 2

3 4 d2,3,4

d3,4,5 5

6 d4,5,7

7d6,8 8 d4,6,7

FIGURE 5.4: A chordal graph and its 1, 2, d1,2-rooted poset.

58 Chapter 5. Finding a maximum-weight convex set in a chordal graph

5.3.2 Reduction to a poset problem

We now give a sufficient condition on a pair (G, K), where G is a chordalgraph and K a maximal clique of G, for the existence of a one-to-one corre-spondence between the non-empty ideals of the K-rooted poset and the dK-rooted convex sets. Given a chordal graph G with clique-separator graph G,for K inKG and a = (S1, S2) in ArG , we say that a is K-blocking if S1 is a mini-mal s2dK-separator for every s2 in S2 \ S1. Figure 5.5 shows a clique-separatorgraph in which (S1, S2) and (S6, S7) are K1-blocking arcs but (S2, S3) and(S5, S4) are not.

K1 S1

K2 S2

K3 S3 K4 S4 K5

S5 K6 S6 K7

S7K8 K9

FIGURE 5.5: A clique-separator graph with exactly two K1-blocking arcs.

As shown in Theorem 5.3.2, the absence of K-blocking arcs is a sufficientcondition for the correspondence between ideals of PK and dK-rooted convexsets. Figure 5.6 below gives a schematic view of the second part of the proof.

Theorem 5.3.2. Let G = (V, E) be a chordal graph with clique-separator graph G,a maximal clique K in KG such that there is no K-blocking arc in G and PK be theK-rooted poset of G. Then a subset I of V is a non-empty ideal of PK if and only if Iis a dK-rooted convex set in G.

Proof. First, let C be a convex set containing dK. For c in C \ dK, any vertexu such that u 6K c belongs to some chordless path. By convexity, we haveu ∈ C so C is an ideal of PK.

Now let I be an ideal of PK and suppose, for contradiction, that I is notconvex. Then by definition, there must exist x and y in I and a chordless path(x, f1, f2, . . . , ft, y) such that f1, f2, . . . , ft do not belong to I. Note that x andy must be incomparable in PK, for otherwise f1 or ft would be contained in I.In particular, they are both different from dK. Moreover we cannot have bothx, dK and y, dK as edges, since otherwise x, y ⊆ K. So without loss ofgenerality, we assume that x, dK 6∈ E.

5.3. A special case solvable in polynomial time 59

Let T be a minimal xy-separator included in the neighborhood N(x) ofx. Let S = T ∩ I. We claim that S is either an xdK-separator or an ydK-separator. Suppose otherwise. Then there must be two chordless paths ofthe form (x, u1, u2, . . . , un, dK) and (y, v1, v2, . . . , vn′ , dK) contained in I andavoiding S. By concatenating them, we obtain a path from x to y in I avoidingS, which contradicts the fact that T was an xy-separator. This proves theclaim.

In fact, S is an xdK-separator because otherwise, we can extract a chordlesspath from (y, ft, . . . , f1, x, . . . , dK) that avoids S and contradicts the fact thatS is an ydK-separator. Now consider a minimal xdK-separator S1 ⊆ S and aminimal vertex separator S2 ⊆ T such that a = (S1, S2) is an arc of G. Weknow such an arc exists because T is an xy-separator while S is not. We nowshow that a is K-blocking, a contradiction.

By definition, a is K-blocking if and only if S1 is a tdK-separator for any t ∈S2 \ S1. Suppose for contradiction that for some such t there exists a chordlesspath from t to dK avoiding S1. We recall that t ∈ T and T ⊆ N(x), hencex, t is in E. But then there is a chordless path from x to dK avoiding S1,contradicting that S1 is an xdK-separator. Hence a is indeed K-blocking.

x y

f1 ft

u1 v1

dK

T

IS

FIGURE 5.6: Illustration of the second part of proof for Theo-rem 5.3.2.

So whenever the graph G has no K-blocking arc, it is possible to computea maximum-weight dK-rooted convex set of G in polynomial time by firstcomputing the cover relation of the K-rooted poset, then using Picard’s algo-rithm [104]. Note that the relation 6K can be computed in polynomial timeas we show later. There are some well-known examples of chordal graphsG such that for every K in KG, the clique-separator graph of G has no K-blocking arc. For example, k-trees have no arc in their clique-separator graph(see Patil [101] for details). A k-tree is a graph formed by starting with a

60 Chapter 5. Finding a maximum-weight convex set in a chordal graph

clique of size k + 1 and then repeatedly adding vertices with exactly k neigh-bors inducing a clique. In the next section, we will see how to deal with thecase where the clique-separator graph contains a K-blocking arc.

5.4 A polynomial-time algorithm

We now consider chordal graphs G with one or more K-blocking arcs in theirclique-separator graph, for some K in KG. We describe an algorithm for find-ing a maximum-weight convex set rooted in K.

For a chordal graph G with clique-separator graph G we define the sub-graph G a for a = (S1, S2) in ArG as the graph induced by the union of S1

and the connected component of G− S1 that intersects S2. Figure 5.7 showsan example of the operation. Note that G a is also a chordal graph (asany induced subgraph of a chordal graph is also chordal).

G = (V, E)

1

2

3 4 5

d2,3,4 d2,4,5

d1,2

G

1, 2, d1,22

2, 3, 4, d2,3,4

2, 4

2, 4, 5, d2,4,5a

H = G a

2

3 4 5

d2,3,4 d2,4,5

H

2, 3, 4, d2,3,4

2, 4

2, 4, 5, d2,4,5

FIGURE 5.7: An example of the operation.

For a chordal graph G = (V, E), a subset R of V and a weight func-tion w, we denote by opt(G, R) a maximum-weight R-rooted convex set ofG with respect to w. If R is a singleton r, we will write opt(G, r) insteadof opt(G, r). The algorithm proceeds in two main steps. In a first pre-processing phase, for each arc a = (S1, S2), we compute opt(G a, S1) thatis, a maximum-weight convex set of G a rooted in the vertex separator

5.4. A polynomial-time algorithm 61

S1. After this preprocessing phase we denote by label(a) the solution of thissub-problem. An algorithm for this preprocessing phase is described in Sec-tion 5.4.2.

5.4.1 Computation phase

In this second phase, we are going to use the labels of the arcs to computea maximum-weight dK-rooted convex. The algorithm proceeds essentiallyby collapsing the vertices of the subgraph (G a) − S1 into a single vertexza for each arc a = (S1, S2) that is K-blocking. The weight of za is then setto w(label(a)) − w(S1), so that the weight of an optimal solution remainsunchanged. This is detailed in Algorithm 1.

Algorithm 1: Finding a maximum dK-rooted convex set in a chordalgraph

Input: a chordal graph G and its clique-separator graph G, a maximalclique K of G, a weight function w, the function label

Output: a maximum-weight K-rooted convex set C1 while ∃ a = (S1, S2) ∈ ArG such that a is K-blocking do2 Identify the vertices of (G a)− S1 into a new vertex za3 w(za)← w(label(a))− w(S1)4 Add a dummy vertex to the new maximal clique za ∪ S15 Update G6 Use Picard’s algorithm to compute a maximal weight dK-rooted convex

set C of G7 Return C

Note that the number of K-blocking arcs decreases at each iteration of theloop. Indeed, at least the vertex separator S2 disappears. One step of the algo-rithm is illustrated by Figure 5.8. Since the goal is to find a maximum-weightconvex set in the graph, we need to remember that including the vertex za ina solution for the collapsed instance amounts to choosing the set label(a) \ S1

in a solution of the original instance.

Theorem 5.4.1. Let G = (V, E) be a chordal graph with a maximal clique K and leta = (S1, S2) be a K-blocking arc of ArG . Let G∗ be the graph obtained from G afterapplying Steps 2–4 of Algorithm 1 on a. Then w(opt(G, dK)) = w(opt(G∗, dK)).

Before proving Theorem 5.4.1, we make two simple observations.

62 Chapter 5. Finding a maximum-weight convex set in a chordal graph

S1

S2

G

S1

S2

a

S1G∗

S1

za, dza∪S1 ∪ S1dza∪S1

za

FIGURE 5.8: Illustration of the transformation in Algorithm 1and the implication for the clique-separator graph.

Lemma 5.4.1. Let G be a chordal graph, S be a minimal vertex separator of G andlet V1 and V2 be the vertex sets of two distinct components of G− S. If C1 and C2 aretwo S-rooted convex sets in the graphs induced by V1 ∪ S and V2 ∪ S respectively,then C1 ∪ C2 is a convex set of G.

Proof. By contradiction, suppose there are c and c′ in C1 ∪ C2 and a chordlesspath (c, f1, . . . , fn, c′) of G with f1, . . . , fn outside of C1 ∪ C2. There must existi in 1, . . . , n such that fi is in S otherwise we have a contradiction with thefact that S is a separator. But then, fi ∈ C1 ∪ C2 because S ⊆ C1 ∩ C2, and wehave a contradiction.

Lemma 5.4.2. Let G = (V, E) be a chordal graph with a maximal clique K and leta = (S1, S2) be a K-blocking arc of ArG . Then, for a dK-rooted convex set C in Gthat contains some vertex of (G a)− S1, we have S1 ⊂ C.

Proof. By contradiction, let s1 be in S1 \ C and let c be in C ∩ (G a − S1).We know that dK and c are not in the same connected component of G− S1.Because a is K-blocking, there is a chordless path (dK, v1, . . . , vn, s1, s2) withs2 ∈ S2 \ S1, s1 ∈ S1 and vn /∈ S1. There is also a path (s1, s′2, . . . , c) in G a with s′2 ∈ S2 \ S1 from which we can extract a chordless path that onlyintersects S1 in s1. So we build a path (dK, v1, . . . , vn, s1, . . . , c) that can nothave a chord, a contradiction.

Proof of Theorem 5.4.1. We decompose the equality into two inequalities. Weshow w(opt(G, dK)) > w(opt(G∗, dK)) first. More precisely, we show thatfor every dK-rooted convex C∗ of G∗, we have a dK-rooted convex set C of Gwith w(C) = w(C∗). If za /∈ C∗ we take C = C∗ and we are done. Now, if

5.4. A polynomial-time algorithm 63

za ∈ C∗, we define C as the union of C∗ \ za (which is convex because N(za)

induces a clique) with label(a). Obviously w(C) = w(C∗), we only need tocheck that C is convex. Because a is K-blocking, N(za) \ dza∪S1

= S1 is aminimal dKza-separator, hence C \ za must include S1. We also know thatS1 is contained in label(a). From Lemma 5.4.1, C is convex in G, thereforew(opt(G, dK)) > w(opt(G∗, dK)).

We now show w(opt(G, dK)) 6 w(opt(G∗, dK)). More precisely, for ev-ery dK-rooted convex C of G, we have a dK-rooted convex set C∗ of G∗ withw(C) 6 w(C∗). If C does not intersect (G a)− S1, we take C∗ = C and weare done. If C does intersect (G a) − S1, we define C∗ as the union of za

with the vertices of C that also are in G∗. We have w(C) 6 w(C∗), otherwisewe contradict the maximality of label(a). From Lemma 5.4.2, the vertices ofC that are also in G∗ form a convex set containing S1. Since za ∪ S1 is aclique, hence is convex, Lemma 5.4.1 implies that C∗ is also convex. So wehave w(opt(G, dK)) 6 w(opt(G∗, dK)).

5.4.2 Preprocessing

We now describe the algorithm for computing the labels for the arcs in G.This step is done only once and does not depend on the root of the con-vex set we are looking for. Recall that the label of an arc a = (S1, S2) is amaximum-weight convex set of G a rooted in S1. Note that this algorithmuses Algorithm 1 as a subroutine on smaller graphs.

The algorithm is composed of two main ingredients. First, we need tolabel the arcs in an order such that the computation only involves arcs thatare already labeled. We prove that we can achieve this by following the orderof inclusion of the graphs G a. Second, in order to compute an optimalconvex set rooted in S1, we need to check all possible roots dK such that S1 iscontained in K. This is detailed in Algorithm 2.

In step 6 of Algorithm 2, we can force S1 to be in the solution C∗ by assign-ing a sufficiently large weight to each vertex of S1 before calling Algorithm 1.More precisely, we assign them the weight ∑v∈V |w(v)|. By Lemma 5.2.2,looking for all the dK-rooted convex sets with K in Ka ensures that we willfind a maximum-weight S1-rooted convex set of G a.

Note that Algorithm 2 labels the arcs in an order compatible with the par-tial order of inclusion of the graphs G a. The following lemma guarantees

64 Chapter 5. Finding a maximum-weight convex set in a chordal graph

Algorithm 2: Labeling the arcs in GInput: a chordal graph G and its clique separator graph G, a maximal

clique K of G, a weight function wOutput: the label function

1 while ∃ an arc in ArG without label do2 Select a = (S1, S2) ∈ ArG without label such that every arc a′ with

G a′ ⊂ G a is already labeled3 M← S14 Let Ka be the set of maximal cliques of G that contain S1 and are

contained in G a5 for K in Ka do6 Using Algorithm 1, compute a maximum-weight convex set C∗

of G a rooted in dK and containing S17 M← maxwM, C∗8 label(a)← M

that the K-blocking arcs that will be processed by Algorithm 1 are all alreadylabeled.

Lemma 5.4.3. Let G be a chordal graph with clique-separator graph G and let a =

(S1, S2) in ArG and a′ = (S3, S4) an arc of the clique-separator graph of G a. IfG a′ 6⊂ G a, then a′ is not K-blocking for any maximal clique K in Ka.

Proof. Suppose that G a′ 6⊂ G a. We show that S3 is not a minimal s4dK-vertex separator for s4 in S4 \ S3. If G a′ = G a, then dK is connected to s4

in (G a′)− S3 and we have the result.If G a′ 6= G a, there is v in G a′ such that v is not in G a. So there

must exist a chordless path p from s4 to v that avoids S3. But, because S4 is inG a and v is not, the path p must contains a vertex s1 ∈ S1. Now, becauses1, dK is an edge in G, we can deduce the existence of a chordless path froms4 to dK that avoids S3.

A complete execution of the algorithm on an example is given in a sectionbelow.

5.5 Analysis

We will now analyze the time complexity of the complete procedure that finda maximum-weight convex set in a chordal graph. A complete example ofthe execution is also provided.

5.5. Analysis 65

S1 S2 S3 S4

v

s1 s4

pdK

FIGURE 5.9: Illustration of the proof of Lemma 5.4.3.

5.5.1 Time complexity

From Algorithms 1 and 2, it seems straightforward that the time complexityneeded to solve the maximum-weight convex set problem on a chordal graphG = (V, E) is bounded by a polynomial in |V| and |E|. More precisely, wesee that the complexity of the algorithm used to solve Problem 4 on G willbe bounded by that of the preprocessing step. Indeed, the preprocessing stepinvolves |V||E| calls to Picard’s algorithm. We recall (from Chapter 4) thatTMClo(n, m) denotes the time complexity for solving the maximum closureproblem for a poset with n elements and m cover relations and TMCut(n, m)

denotes the time complexity for solving the minimum s, t-cut, in a s, t-graphwith n vertices and m arcs. We have TMClo(n, m) 6 O(TMCut(n, m)) (seePicard [104]) and TMCut(n, m) 6 O(nm) (see Orlin [98]). Hence the overallrunning time of our algorithm is O(|V|2|E|2). If we denote by n the numberof vertices of the input graph, then this is bounded by O(n6).

To prove that the problem can be solved in this running time we need toshow that all the information we need in Algorithms 1 and 2 can be com-puted in a time bounded asymptotically by the time of the preprocessingstep. More precisely, given the chordal graph G, we can compute the follow-ing information in O(|V|2|E|2) time: the clique-separator graph G of G, thevertices in G a for each a in ArG , the cliques inKG for which a is K-blockingfor each a in ArG , the clique inKa for each a in ArG , the matrix of the relations6K for each K in KG and a total order on the arcs G such that G a ⊆ G a′

implies a < a′ for a, a′ in ArG . The detailed proofs are given below. Thisconcludes the proof of Theorem 5.2.1. Note that the algorithm for solving theproblem on split graphs (from Chapter 4) has a better time complexity, butcannot be applied to all chordal graphs.

We now prove that for a chordal graph G = (V, E) given by its adja-cency matrix, and a weight function on the vertices, the running time of our

66 Chapter 5. Finding a maximum-weight convex set in a chordal graph

algorithm is in O(|V|2|E|2), in the worst-case. We need to check that the in-formation needed for the execution of Algorithms 1 and 2 can be computedin advance, only once for a given graph. For a graph G, finding a connectedcomponent of G−X that contains v where X ⊆ V and v ∈ V \X can be donein time O(|V|+ |E|) as stated by Hopcroft and Tarjan [68]. We will use thefollowing lemma.

Lemma 5.5.1. Given a (connected) chordal graph G = (V, E) with clique-separatorgraph G, we have |V| − 1 6 |E| and |ArG | 6 |E|.

Proof. The first inequality follows from connectedness. For the second in-equality, notice each arc (S1, S2) ∈ ArG is generated by two cliques S1 and S2

such that S1 ⊂ S2 and there is no vertex S3 such that S1 ⊂ S3 ⊂ S2. We assignto each arc a = (S1, S2) a unique edge s1, s2 in G with s1 ∈ S1 and s2 ∈ S2.This implies |ArG | 6 |E|.

Lemma 5.5.2. Given a chordal graph G = (V, E) and its clique-separator graph G,listing G a for all a in ArG can be done in O(|E|2) time.

Proof. For each arc a = (S1, S2), we need to find the connected componentof G − S1 that intersects S2. Using Lemma 5.5.1, we have O(|E|2) as totalrunning time.

Lemma 5.5.3. Given a chordal graph G = (V, E) with clique-separator graph G,obtaining the list, for each a in ArG , of the cliques in KG for which a is K-blockingtakes O(|E|2|V|2) time.

Proof. By Lemma 5.5.2 we can obtain the list of G a for all a in ArG in O(|E|2)time. Then, for each a = (S1, S2) in ArG , and for each K in KG, we check twoconditions. First, that dK is not in G a (i.e. S1 is a dks2-separator for s2 inS2). Second, that there is no a′ = (S3, S1) in ArG such that dK is not in G a′

(i.e. S1 is minimal among the s2dK-separator for every s2 in S2).

Lemma 5.5.4. Given a chordal graph G = (V, E) and its clique-separator graph G,obtaining the relations 6K for all K in KG takes time O(|V|2|E|2).

Proof. By Lemmas 5.5.2 and 5.5.3 we can obtain, for each arc a in ArG , thevertices in G a and the cliques in KG for which a is K-blocking in ArG inO(|E|2|V|2). First, for all K in KG, we set dk 6K k for all k in K. Second, forall K in KG, and for all a = (S1, S2) in ArG such that a is K-blocking, we sets 6K u for all s ∈ S1 and u ∈ (G a)− S1.

5.5. Analysis 67

Lemma 5.5.5. Given a chordal graph G = (V, E) and its clique-separator graph G,sorting the arcs of ArG such that G a ⊆ G a′ implies a < a′ can be done inO(|V||E|2) time.

Proof. We use Lemma 5.5.2 to obtain a list of G a for all a in ArG . Thecomparison between G a and G a′ for a, a′ in ArG takes O(|V|) time.Hence sorting takes O(|V||E|2) time.

Lemma 5.5.6. Given a chordal graph G = (V, E) with clique-separator graph Gand the list of G a for all a in ArG , obtaining elements in Ka for all a in ArG takesO(|E||V|2) time.

Proof. For each arc a = (S1, S2) in ArG , we look at each K ∈ KG such thatS1 ⊂ K, and we check if the clique is in G a.

Finally, we have the main result of the section.

Theorem 5.5.1. Given a chordal graph G = (V, E) (encoded an adjacency matrix),a maximum-weight convex set in the monophonically convex geometry defined on Gcan be found in O(|V|2|E|2) time.

5.5.2 Detailed example

Looking at Algorithms 1 and 2, it seems possible to merge them to save com-putation time. But the situation is not that simple, because in Theorem 5.4.1,the assumption that a is K-blocking cannot be removed. To illustrate themechanism of the two algorithms, we give an example. Figure 5.10 showsa chordal graph G, its clique-separator graph G and a weight function. Thegoal is to compute opt(G, dK1).

We look at the preprocessing phase first, and we use Algorithm 2 to labela1 and a2. Because G a2 ⊂ G a1, we begin by computing label(a2). Inother words, we want to compute a maximum-weight S2-rooted convex setof G a2, illustrated in Figure 5.11. There is no K-blocking arc in G a2 forany K in Ka2 , we can directly use Picard’s algorithm on each clique in Ka2

after temporarily changing the weight of the vertices in S2 in order to imposethat S2 be contained in the solution. So Picard’s algorithm is used twice, withK4 and K5, and we keep the best solution among the outputs. The result willbe a convex set of weight 1, for instance 2, 4, 6, 7, 8, dK5, which becomeslabel(a2).

68 Chapter 5. Finding a maximum-weight convex set in a chordal graph

11

20

31

4−4

5−1

64

7−2

83

dK2

0

dK3

0

dK1

0

dK4

0

dK5

0

K1 S1

S2

S3

K2 K3

K4 K5

a1

a2

FIGURE 5.10: A chordal graph and its clique-separator graphfor illustrating Algorithms 1 and 2.

20

4−4

64

7−2

83

dK4

0

dK5

0

S2S3K4 K5

FIGURE 5.11: The graph G a2 and its clique-separator graph.

Now we compute label(a1), so we are looking for an S1-rooted convex setin G a1 represented in Figure 5.12. We temporarily change the weight of thevertices in S1 and we run Algorithm 1 on the graph G a1 four times, withthe cliques K2, K3, K4 and K5. For the clique K4 there is no K4-blocking arcand we use Picard’s algorithm. The same process is applied with K5. For theclique K2, the arc a2 is K2-blocking but already labeled. So we identify the ver-tices of (G a2)− S2 to a vertex za2 with weight w(label(a2))− w(S2) = 1.After this operation, there is no K2-blocking arc and we use Picard’s algo-rithm. Figure 5.13 shows a visual representation of the transformation. Thesame process is applied with K3. For the labeling of a1 we have used Picard’salgorithm four times. A best S1-rooted convex set of G a1 is 2, 6, dK4withweight 4.

Now every arc is labeled, and we look at the computing phase. Becausewe want a maximum-weight dK1-rooted convex set, we run Algorithm 1.

5.5. Analysis 69

20

31

4−4

5−1

64

7−2

83

dK2

0

dK3

0

dK4

0

dK5

0

S2

S3

K2 K3

K4 K5a2

S1

FIGURE 5.12: The graph G a1 and its clique-separator graph.

20

31

4−4

5−1

dK6

0

za2

1

dK2

0

dK3

0 S2

dK6 , za2 ∪ S2

K2 K3S1

FIGURE 5.13: The graph after the identification of vertices inorder to remove K2-blocking arcs.

There is only one K1-blocking arc, namely a1. So we identify the vertices of(G a1)− S1 to a vertex za1 with weight w(label(a1))−w(S1) = 4. After thisoperation, there is no K1-blocking arc as shown in Figure 5.14, and we use Pi-card’s algorithm to find 1, 2, za1 , dK1 as a maximum-weight K1-rooted con-vex set, which gives rise to the convex set C∗ = 1, 2, dK1∪ (label(a1) \ S1) =

1, 2, 6, dK1 , dK4 of G, with w(C∗) = 5.

11

20

za1

4

dK1

0

dK7

0 K1 S1 2, za1 , dK7

FIGURE 5.14: The state of the graph after removing the K1blocking arc.

71

Chapter 6

The realizability problem forconvex geometries

“We’ve always defined ourselves by the ability to overcome the impossible. Andwe count these moments. These moments when we dare to aim higher, to breakbarriers, to reach for the stars, to make the unknown known.”

— Joseph Cooper in Interstellar

In this chapter we discuss the realizability problem for convex geome-tries. After providing some context about the problem, we use a hardnessresult for the realizability of abstract order types to show that deciding therealizability of an affine convex geometry in the plane is ∃R-hard (hence alsoNP-hard). This is related to a problem posed by Edelman and Jamison [42]in their seminal paper. The results in this chapter were obtained in collabora-tion with Hoffmann [67]. Originally, the hardness result for the realizabilityproblem for convex geometries was based on results involving “allowablesequences”. We give here a simpler version of the proof only involving ab-stract order types. Appendix A provides basic definitions and properties ofallowable sequence and a summary of the original proof. For details andcontext regarding the computational geometry concepts used in this chapter,see Goodman et al. [55] and Richter-Gebert [110].

6.1 Basics of computational geometry

We first introduce some key concepts in computational geometry and definethe realizability problem in different contexts. We also introduce the existen-tial theory of the reals.

72 Chapter 6. The realizability problem for convex geometries

6.1.1 Affine convex geometries

We know that starting with a finite set of points P in Rd, we obtain an affineconvex geometry (P, CP) with CP equal to C ⊆ 2P : conv(C) ∩ P = C. Fora convex geometry (V, C) we say that (V, C) is realized by a set of points Pin Rd if (V, C) is isomorphic to (P, CP). Edelman and Jamison [42] posed theproblem of characterizing the convex geometries that can be realized by afinite point set in Rd. We will, in a first time, mainly focus on affine convexgeometries induced by point sets in R2 and show that even in this specialcase, the problem is hard. We define the realizability problem for convexgeometries as the following decision problem. Given a copoint poset of aconvex geometry (V, C), is there a point set P in R2 such that the affine con-vex geometry induced by P is isomorphic to (V, C)? Here the notion of iso-morphism between two convex geometries (V1, C1) and (V2, C2) means thereis a bijection f : V1 → V2 such that for all C in 2V1 we have C ∈ C1 if andonly if f (C) ∈ C2. Note that an affine convex geometry (P, C) has at most|P|2 copoints. Indeed, the convex sets for affine convex geometries are de-fined using intersections of half-spaces spanned by two points of P. Becausecopoints are the convex sets that are not the intersection of other convex sets,the convex geometry has at most |P|2 copoints.

Adaricheva and Wild [5] show some relation between special cases ofaffine convex geometries and order types, implying that the computationalproblem of deciding a modified version of the Edelman–Jamison problem ishard.

We mention some results connected to the realizability problem for con-vex geometries. Kashiwabara et al. [75] showed that any convex geometrycan be represented as a generalized shelling in Rd for some d. Richters andRogers [109] reproved this theorem giving a better bound on the dimension d.Different representations of finite convex geometries using different shapesthan points for the elements of the ground set have been studied, see forinstance Adaricheva and Bolat [3], Czedli [30] or Czedli and Kincses [31].

In the following sections, we give a simple construction showing that de-ciding the realizability of convex geometries in R2 is NP-hard (in fact weprove a stronger result). This construction can be lifted in any dimension,thus we obtain a hardness result for the original Edelman–Jamison problem.Formally, we reduce an ∃R-complete realizability problem for abstract or-der types to the realizability problem for affine convex geometries. All these

6.1. Basics of computational geometry 73

notions are introduced below.

6.1.2 Abstract order types and chirotopes

Combinatorial abstractions of properties of point sets in Rd are useful tools incomputational geometry. For example, many algorithms on point sets onlyrequire the “order type” of the point instead of the exact coordinates, whichmakes many algorithms in computational geometry purely combinatorial.For a point set P in the plane, the chirotope associated with P is the mappingϕ : P3 → −1, 0, 1, where

ϕ(p1, p2, p3) = sgn

det

1 1 1p1x p2x p3x

p1y p2y p3y

.

The map ϕ encodes an orientation (clockwise, collinear, or counterclockwise)for triples of points. The other way of seeing a chirotope is that ϕ(p1, p2, p3)

gives us the information of the position of p2 with respect to the directed linethrough (p1, p3). The information can be “to the left of”, “to the right of”or “contained in” the line. For a point set P in the plane, the pair (P, ϕ) isan order type if ϕ is the chirotope of P. Figure 6.1 shows a point set andthe order type associated with it. By the properties of the determinant in thedefinition of ϕ, the four equalities ϕ(a, b, c) = 1, ϕ(a, b, d) = −1, ϕ(a, c, d) = 1and ϕ(b, c, d) = 0 completely determine the order type.

a

b c d

ϕ(a, d, b) = −1 ϕ(a, b, c) = 1 ϕ(a, c, d) = 1

ϕ(b, c, d) = 0

FIGURE 6.1: Illustration of an order type.

For a finite set E, the pair (E, χ) is an abstract order type if χ : E3 →−1, 0, 1 satisfies the four acyclic rank 3 chirotope axioms which are thefollowing:

74 Chapter 6. The realizability problem for convex geometries

• χ is not identically zero,

• χ(pρ(1), pρ(2), pρ(3)) = sgn(ρ)χ(p1, p2, p3) for p1, p2, p3 in E and any per-mutation ρ,

• there are p1 and p2 in E such that χ(p1, p2, q1) > 0 for all q1 ∈ E,

• if χ(q1, p2, p3)χ(p1, q2, q3) 6= 1 and χ(q2, p2, p3)χ(q1, p1, q3) 6= 1 andχ(q3, p2, p3)χ(q1, q2, p1) 6= 1, then χ(p1, p2, p3)χ(q1, q2, q3) 6= 1, for anyp1, p2, p3, q1, q2, q3 in E.

More details and context are given in Bjorner et al. [15]. Obviously, theorder type induced by a point set in R2 is an abstract order type. But abstractorder types are strict abstractions of point sets, hence not all abstract ordertypes can be represented as a set of points in the plane. An abstract ordertype (E, χ) is realizable if there exists a point set in the plane with chirotopeχ. This leads to the realizability problem for abstract order types: givenan abstract order type (E, χ), decide if there exists a planar point set P, theorder type of which is isomorphic to (E, χ). We say that two abstract ordertype (E1, χ1) and (E2, χ2) are isomorphic if there is a bijection f : E1 →E2 such that for all a, b, c in E1 we have χ1(a, b, c) = χ2( f (a), f (b), f (c)).The smallest example of a non-realizable abstract order type is given by themodified Pappus configuration illustrated in Figure 6.2. The orientations inthe abstract order type are given by the lines and one curve. So e is on theright side of the directed line through (d, f ) (i.e. the orientation of (d, e, f )is counterclockwise). But in R2, the points d, e and f must be collinear byPappus’s hexagon theorem. For more details on Pappus configuration, seeGrunbaum [62].

ab

c

d e f

g h i

FIGURE 6.2: A non-realizable modified Pappus configuration.

6.1. Basics of computational geometry 75

An abstract order type induces an AOT convex geometry by definition ofthe latter (see Chapter 3).

Lemma 6.1.1. An abstract order type uniquely determines an AOT convex geome-try.

But a given AOT convex geometry can be induced by several distinct ab-stract order types. This is illustrated by Figure 6.3 where two distinct (real-izable) abstract order types on P = a, b, c, d induce the convex geometry(P, 2P). For an AOT convex geometry (V, C) we say that an abstract ordertype is a representation of (V, C) if the abstract order type induces (V, C).Obviously an AOT convex geometry can have more than one representation.

CP = 2P

χ(a, b, c) = 1

a

b

CP = 2P

χ(a, b, c) = 1χ(a, d, c) = 1 χ(a, d, c) = −1

c

d

a

b

d

c

FIGURE 6.3: Two abstract order types inducing the same AOTconvex geometry.

The concept of (abstract) order types appears under different names inthe literature, for example in Knuth [80] where alternative axioms for acyclicuniform rank 3 oriented matroids are studied under the name of CC-systems.Another important result is the Topological Representation Theorem due toFolkman and Lawrence [51] using pseudoline arrangements which are ar-rangements of oriented x-monotone curves that pairwise intersect exactlyonce in the Euclidean plane. The theorem shows that every abstract ordertype can be represented as an arrangement of oriented pseudolines. SeeBokowski [17] for the detail of a proof.

It is a natural question to ask how the point sets that agree with the com-binatorial description looks like. This question leads to the following notion.For an abstract order type (E, χ) with |E| = n, the realization space of (E, χ)

is the set of all points in R2n that gives a realization of (E, χ) in R2 where apoint set with n elements in R2 is naturally mapped to a point in R2n. In hisfamous Universality Theorem, Mnev [93] proved that for any arbitrary semi-algebraic set S there is some abstract order type whose realization space has,in some sens, the same structure as S

76 Chapter 6. The realizability problem for convex geometries

6.1.3 Existential theory of the reals

To provide a hardness result for the realizability problem for affine convexgeometries, we will use the existential theory of the reals. In terms frommathematical logic, the theory consists in the set of all true sentences of theform

∃x1, . . . , ∃xn, F(x1, . . . , xn),

where F(x1, . . . , xn) is a quantifier-free formula involving equalities and in-equalities of real polynomials connected by “and” connectors. The existentialtheory of the reals leads to the following decision problem: given a systemof polynomial equalities and inequalities, decide if there is an assignment ofreal values to the variables such that the system is satisfied. Using this prob-lem as “complete” we define a complexity class called ∃R. In other word, adecision problem P is in ∃R if there is a polynomial reduction to P from thedecision problem induced by the existential theory of the reals. See Basu etal. [11] for more details. The only known relations to other complexity classesareNP ⊆ ∃R ⊆ PSPACE, where PSPACE is the class of decision problemssolvable by a Turing machine in polynomial space. The second relation is aresult by Canny [22]. Whether one of the relations is strict or not is unknown.

The complexity class ∃R has been introduced by Schaefer [113], moti-vated by the continuously expanding list of (geometric) problems that havebeen shown to be complete in this class. One of the first geometric ∃R-complete problems is the abstract order type realizability problem. The re-duction is due to Mnev with his Universality Theorem: for each polynomialinequality system S that consists of strict inequalities and equalities there isan abstract order type whose realization space is “stably equivalent” to thegeometric set described by S. This means that the abstract order type con-structed from S is not only realizable if the set solutions of S is non-empty,but the set of solutions also has the same “structure”. The structures pre-served by the equivalence relation include the homotopy group and the al-gebraic complexity of the numbers (e.g., “does S have a rational solution”),see Richter-Gebert and Ziegler [111] for more consequences of stable equiv-alence. We give a more detailed introduction about this concept in the nextsection.

Many geometric problems can be shown to be ∃R-hard by a reduction

6.2. Hardness result for the realizability problem 77

from realizability problem for abstract order types (or the dual problem in-volving pseudoline arrangements), see Goodman et al. [55] for details. Some∃R-complete geometric problems include recognition of intersection graphsof segments (see Kratochvıl and Matousek [85]), intersection graphs of discs(see McDiarmid and Muller [92]), intersection graphs of convex sets (seeSchaefer [113]) and point visibility graphs (see Cardinal and Hoffmann [26]),the art gallery problem (see Abrahamsen et al. [1]), realizability of the face lat-tice of a 4-polytope (see Richter-Gebert and Ziegler [111]) and d-dimensionalDelaunay triangulations (see Adiprasito et al. [6]), computing the rectilin-ear crossing number (see Bienstock [13]) and planar slope number (see Hoff-mann [66]). Recently, Roy [112] published an ingenious and rather involvedNP-hardness proof for recognition of arbitrary point visibility graphs. Werefer to Cardinal [23] for an overview.

6.2 Hardness result for the realizability problem

In this section, we show that the realizability problem for convex geometries,encoded by a copoint poset, is ∃R-hard. First, we state some technical re-sults on abstract order types and AOT convex geometries. Next, we define aspecific family of abstract order types, which are the building blocks for ourmain result. Finally, we reduce the realizability problem for abstract ordertypes to the realizability problem for convex geometries.

6.2.1 Overview

The main result of this chapter is the following theorem.

Theorem 6.2.1. The realizability of convex geometries encoded by a copoint poset is∃R-hard.

However, before giving a proof for Theorem 6.2.1, we show a closely re-lated result.

Theorem 6.2.2. The realizability of AOT convex geometries, encoded by an abstractorder type, is ∃R-hard.

The main idea of the proof for Theorem 6.2.2 is to show that, for somespecific AOT convex geometries with additional information about their “ex-treme points”, we can retrieve the “only” abstract order type that induces the

78 Chapter 6. The realizability problem for convex geometries

convex geometry. The technical results presented below are useful to definethis process. The additional information about the extreme points will begiven by the construction of a “fixed ring,” which can be viewed as furtherpoints added to the abstract order type to force a specific counterclockwisesequence of extreme points in every (potential) realization of the abstract or-der type.

6.2.2 Technical properties

We extend the notion of extreme points to abstract order types. For an ab-stract order type (P, χ), a point p in P is extreme if p is extreme in the AOTconvex geometry induced by (P, χ). The definition is equivalent to the one inLas Vergnas [86]. The following lemma is a direct consequence of the defini-tion of extreme points and the application of the Jarvis March (see Jarvis [73]).Jarvis March is used to detect the extreme points of a convex hull from a givenpoint set in the plan. Starting from a rightmost point of the point set, we keepthe points in the convex hull by counterclockwise rotation. From a currentpoint, we can choose the next point by checking the orientations of thosepoints from the current point. When the angle is largest, the point is chosen.This process can directly be adapted to abstract order types. Figure 6.4 givesan illustration. Proofs and context can be found in Goodman et al. [55].

Lemma 6.2.1. For any abstract order type (P, χ) with an extreme point r1, there isa unique sequence of extreme points (r1, . . . , rk) such that r1, . . . , rk are in gen-eral position, and for i in 1, . . . , k, χ(ri, ri+1, a) > 0 for every a in P, with theconvention that rk+1 is r1.

r1

r2

r3

FIGURE 6.4: Illustration of Jarvis March.

6.2. Hardness result for the realizability problem 79

Note that the application of Jarvis March is polynomial. Given an AOTconvex geometry induced by (P, χ) with an extreme point r1, we call the(counterclockwise) extreme cycle the unique sequence from Lemma 6.2.1.In the following, when we mention an extreme cycle (r1, . . . , rk) and performsome operations on the point ri, we assume for convenience that an index ilarger than k is identified with an element of 1, . . . , k as follow. The index isidentified with k if i is a multiple of k and with i mod k otherwise. A directconsequence of the previous lemma is the following corollary (also a directconsequence of Jarvis March [73] for abstract order type).

Corollary 6.2.1. For any abstract order type (P, χ) with a, b in P, r1 an extremepoint and (r1, . . . , rk) the unique counterclockwise sequence of extreme points E, thesets Ra,b,1 = r ∈ E : χ(a, r, b) = 1 and Ra,b,−1 = r ∈ E : χ(a, r, b) = −1 arecomposed of consecutive subsequences of (r1, . . . , rk). Moreover, there are at mosttwo elements r in E such that χ(a, r, b) = 0.

In the following, for a convex geometry (P, C) we denote by τC the clo-sure operator associated to it. The next lemma is a direct consequence ofthe definition of convex sets for AOT convex geometry. Figure 6.5 gives anillustration of Lemma 6.2.2.

Lemma 6.2.2. Let (P, χ) be an abstract order type that induces an AOT convexgeometry (P, C) and a, b, c, d in P in general position. We have that d is inτC(a, b, c) if and only if χ(a, d, b) = χ(b, d, c) = χ(c, d, a).

Proof. Directly, d in τC(a, b, c) means that every convex set containing a, band c also contains d. So we must have χ(a, d, b) = χ(a, c, b) and χ(b, d, c) =χ(b, a, c) and χ(c, d, a) = χ(c, b, a), otherwise, using the definition of convexsets we build a convex set containing a, b and c, but not d. Using the acyclicrank 3 chirotope axioms (from Subsection 6.1.2) we have the result.

Conversely by contradiction, if d is not in τC(a, b, c) then χ(a, d, b) 6=χ(a, c, b) or χ(a, d, c) 6= χ(a, b, c) or χ(b, d, c) 6= χ(b, a, c). Suppose we haveχ(a, d, b) 6= χ(a, c, b), then χ(a, d, b) 6= χ(b, d, c). The other cases are similar.

The next lemma highlights a connection with a given pair of points (a, b)and a given element of the extreme cycle. The two cases of the proof areillustrated by Figure 6.6.

80 Chapter 6. The realizability problem for convex geometries

a

cb

d

a

cb

d

FIGURE 6.5: Illustration of the relation between orientationsand the convex closure.

Lemma 6.2.3. Let (P, χ) be an abstract order type and r1 an extreme point such that(r1, . . . , rk) is the extreme cycle E. Let (P, C) be the induced AOT convex geometryand a, b in P with a 6= b. There is exactly one point ri in E such that a is not inτC(ri+1, b) and a is in τC(ri, ri+1, b).

Proof. First, we suppose that a is in τC(rj, rj+1) \ rj for rj in E. in thatcase we take rj = ri if b /∈ τC(a, rj) and rj+1 = ri otherwise. So in thefollowing we suppose that a is not in τC(rj, rj+1) \ rj for a rj in E. Second,note that for a points rj in E, if b is in τC(rj, rj+1) \ rj then a is outsideτC(rj, rj+1, b). So now we consider rj in E such that χ(rj, rj+1, b) 6= 0. If rj

is such that χ(a, b, rj) = χ(a, b, rj+1) 6= 0 , then a, b, rj and rj+1 are in generalposition and by Lemma 6.2.2, a is outside τC(rj, rj+1, b).

So by Corollary 6.2.1, there are only two pairs of candidates ri, ri+1 forthe property that a is in the set τC(ri, ri+1, b), because ri and ri+1 cannot beboth in Ra,b,1 or both in Ra,b,−1.

Suppose ri is not collinear with a and b, such that χ(a, b, ri) 6= χ(a, b, ri+1).There are two cases to consider: χ(ri+1, b, a) = 1 and χ(ri+1, b, a) = −1.We know that χ(ri, b, ri+1) = 1 so if χ(ri+1, b, a) = 1, then, using acyclicrank 3 chirotope axioms (from Subsection 6.1.2) and Lemma 6.2.2, b is inτC(ri, ri+1, b) and a is not in τC(ri, ri+1, b). Now if χ(ri+1, b, a) = −1,then a is in τC(ri, ri+1, b). If there is rj in E such that a is in τC(rj, b), thenwe have that a is in τC(rj, rj+1, b). Note that a is also in τC(rj, rj−1, b), butrj−1, rj is not a valid candidate because a is in τC(rj, b).

The next lemma is more technical and shows how the orientation of atriple (a, b, c) in an abstract order type imposes some constraints to the con-vex closure in the AOT convex geometry. Figure 6.7 shows in R2 the threepossible cases.

6.2. Hardness result for the realizability problem 81

a

b

ri+1 ri ri+1 ri

a

b

FIGURE 6.6: Two different cases for the selection of (ri, ri+1) inthe proof of Lemma 6.2.3.

Lemma 6.2.4. Let (P, χ) be an abstract order type and r1 an extreme point suchthat (r1, . . . , rk) is the extreme cycle E. Let (P, C) be the induced AOT convexgeometry and a, b in P and c in P \ E such that a and b are not both collinearwith rl and rl+1 for l in 1, . . . , k, and χ(a, c, b) = 1. Let ri and rj in E besuch that a is in τC(ri−1, ri, b) and b is in τC(rj, rj+1, a) with a outside ofτC(b, ri−1) and b outside of τC(a, rj). Then, either c is in τC(a, b, ri, . . . rj)or a is in τC(b, c, ri−1) or b is in τC(a, c, rj+1).

Proof. First, suppose ri and rj+1 are collinear with a and b. Then, c must be inτC(a, b, ri, . . . rj+1). Second, suppose that ri and rj+1 are both not collinearwith a and b. If c is not in τC(a, b, ri+1, . . . rj), then c must be in τC(ri−1, ri, a)or τC(rj, rj+1, b). Consider the first case: c in τC(ri−1, ri, a). We show that a is inτC(b, c, ri−1). The fact that c is in τC(ri, ri+1, a) and a is in τC(ri, ri+1, b) canbe translated, using Lemma 6.2.2, by χ(ri, c, ri+1) = χ(ri+1, c, a) = χ(a, c, ri)

and χ(ri, a, ri+1) = χ(ri+1, a, b) = χ(b, a, ri).Moreover χ(ri−1, c, ri) = χ(ri−1, a, ri) = 1 and χ(a, c, b) = 1 by hypoth-

esis. This means that χ(ri−1, a, c) = χ(c, a, b) = χ1(b, a, ri−1) = −1 or,again using Lemma 6.2.2, a is in τ(b, c, ri−1). In a similar fashion, c inτC(rj, rj+1, b) leads to the fact that b in τ(a, c, rj+1).

a

b

ri ri−1

rj rj+1

c a

b

ri ri−1

rj rj+1

c

a

b

ri ri−1

rj rj+1

c

FIGURE 6.7: The three possibles cases of Lemma 6.2.4.

82 Chapter 6. The realizability problem for convex geometries

In the next two results, we compare two abstract order types (P, χ1) and(P, χ2), which induce the same convex geometry. Formally, this means thatevery subset of P is a convex set in the convex geometry induced by (P, χ1)

if and only if it is a convex set in the convex geometry induced by (P, χ2).The equality between the two abstract order types in Proposition 6.2.1 willbe useful in the reduction presented in Subsection 6.2.4.

Lemma 6.2.5. Let (P, χ1) and (P, χ2) be two abstract order types that both havean extreme point r1. If the two abstract order types have the same counterclockwisesequence of extreme points (r1, . . . , rk) and if they induce the same convex geometry,then χ1(a, r, b) = χ2(a, r, b) for any extreme point r and every a, b in P \ r.

Proof. Consider two abstract order types (P, χ1) and (P, χ2) with the samecounterclockwise sequence of extreme points (r1, . . . , rk) such that they in-duce the same convex geometry (P, C). We denote by E the set of extremepoints. By Lemma 6.2.3, there is exactly one point ri in E such that a is notin τC(ri+1, b) and a is in τC(ri, ri+1, b). Moreover, there is exactly one pointrj in E such that b is not in τC(rj+1, a) and b is in τC(rj, rj+1, a). Using Corol-lary 6.2.1, Lemma 6.2.2 and the fact that the counterclockwise sequence ofextreme points is fixed, we have the result.

Proposition 6.2.1. Let (P, χ1) and (P, χ2) two abstract order types that both havean extreme point r1. If the two abstract order types have the same extreme cycle(r1, . . . , rk) and if they induce the same convex geometry, then χ1 = χ2.

Proof. Consider two abstract order types (P, χ1) and (P, χ2) that both have anextreme point r1 with the same counterclockwise sequence of extreme points(r1, . . . , rk). Also suppose that (P, χ1) and (P, χ2) induce the same convexgeometry (P, C). We call E the set of extreme points. Note that if there existsthree points that are collinear in one abstract order type and not in the other,then the induced convex geometries are not equal. We suppose that χ1 6= χ2

and we show a contradiction. Because χ1 6= χ2, there are a, b and c in P, suchthat χ1(a, c, b) = 1 and χ2(a, c, b) = −1. By Lemma 6.2.5, the three pointscannot all be extreme points, so we suppose that c is not in E. Moreover, aand b cannot be such that a, b ⊂ τC(rl, rl+1) for l in 1, . . . , k, otherwiseχ1(rl, rl+1, c) = −1 or χ2(rl, rl+1, c) = −1, which is not compatible with thedefinition of the extreme cycle.

By Lemma 6.2.3, there are two unique ri and rj in E such that a is inτC(ri−1, ri, b) and b is in τC(rj, rj+1, a) with a outside of τC(b, ri−1) and

6.2. Hardness result for the realizability problem 83

b outside of τC(a, rj). Because χ1(a, c, b) = 1, Lemma 6.2.4 gives us that ei-ther c is in τC(a, b, ri, . . . rj) or a is in τC(b, c, ri−1) or b is in τC(a, c, rj+1).

Note that C = p ∈ P : χ2(a, p, b) > 0 and C′ = p ∈ P : χ2(a, p, b) >0 must be in C. The case c in τC(a, b, ri, . . . rj) is not valid because theintersection between τC(a, b, ri, . . . rj) and C is a convex set that includesa, b, ri, . . . rj, but not c. Again a in τC(b, c, ri−1) is also not valid becauseC′ ∪ b is a convex set and τC(b, c, ri−1) ∩ (C′ ∪ b) includes b, c, ri−1but not a. Similarly, b is in τC(a, c, rj+1) leads to a contradiction becauseτC(a, c, rj+1) ∩ (C′ ∪ a) includes a, c, rj+1 but not b.

6.2.3 The fixed ring

We proceed to define abstract order types, which are the building blocks forour reduction. We define Dk as the (abstract) order type of the followingpoint set in R2. We suppose that k > 2. Consider a (convex) k-gon withvertices (r1, . . . , rk) in counterclockwise order. Then add k points r′1, . . . , r′k,one on each edge of the k-gon, where r′i is placed on the edge induced by ri

and ri+1 with i ∈ 1, . . . , k − 1 and r′k is placed on the edge induced by rk

and r1. For convenience, we use the convention that ri = r(i mod k)+1 and r′i =r′(i mod k)+1. We call Dk a fixed ring and denote the convex geometry inducedby this abstract order type by (Dk, Ck). Note that the only collinearities in Dk

appear among the triples ri, r′i and ri+1. This means we obtain the sameabstract order type, regardless of the k-gon chosen in the construction, asillustrated by Figure 6.8.

r1r2

r3

r4 r5

r6

r′1r′2

r′3

r′4

r′5

r′6 r1

r2

r′3

r′4

r′5r′6

r′1

r′2

r′3

r′4r′5

r′6

FIGURE 6.8: Two point sets inducing D6.

Lemma 6.2.6. In (Dk, Ck), for a triple of extreme points (ra, rb, rc), the point r′i isin the convex closure of ra, rb, rc if and only if ri, ri+1 ⊂ ra, rb, rc.

84 Chapter 6. The realizability problem for convex geometries

Proof. First, suppose that r′i is in the convex closure of ra, rb, rc. We musthave ri, ri+1 ⊂ ra, rb, rc because Dk \ ri and Dk \ ri+1 are in Ck. Sec-ond, if ri, ri+1 ⊂ ra, rb, rc then r′i is in the convex closure of ra, rb, rcbecause r′i is between ri and ri+1.

The next lemma shows why we call Dk the fixed ring. Figure 6.9 showsthe situation in the proof.

Lemma 6.2.7. In each realization Q of (Dk, Ck) the order of the extreme points onthe convex hull of Q is (r1, . . . , rk) or (rk, . . . , r1).

Proof. Suppose there is a realization of (Dk, Ck) with a different order of ex-treme points on the convex hull. Suppose that rx and ry are the two neigh-bors or ri in the extreme point order, with none of x and y equal to i + 1. ByLemma 6.2.6, r′i must be outside of conv(ri, rx, ry) and conv(ri+1, rx, ry).Then we have a contradiction, because r′i must be in conv(ri, ri+1) by defi-nition and conv(ri, ri+1) is contained in conv(ri, ri+1, rx, ry).

ri ri+1

rx

ry

r′i

FIGURE 6.9: The contradiction in the proof of Lemma 6.2.7.

6.2.4 The reduction

We prove Theorem 6.2.1 below. We set up a reduction from realizability ofabstract order type. So given an abstract order type (P, χ), we build anotherabstract order type (P′, χ′), which induces an AOT convex geometry (P′, C).Then we show that (P′, C) is realizable if and only if (P, χ) is realizable. Theabstract order type (P′, χ′) will be called the augmented order type of (P, χ),a formal definition is given below. Roughly speaking, we merge a fixed ringDk to P, where k is the number of extreme points of (P, χ). Given an extreme

6.2. Hardness result for the realizability problem 85

point r of (P, χ), we merge r with r1 and merge the extreme cycle of Dk (start-ing with r1) with the extreme cycle of (P, χ) (starting with r). Then we addthe points r′i, “very close” to the points ri on the “outer layers” defined bythe extreme points. So P′ = P ∪ r′1, . . . , r′k. The extreme points of the AOTconvex geometry will be referred to as E = r1, . . . , rk.

Formally, to complete the definition of χ′, we have to define the orienta-tion of triples containing two points of P and one point of r′1, . . . , r′k andvice versa. Figure 6.10 gives an illustration of an augmented order type.

First, for pa, pb in P and r′i in E, we have

χ′(pa, r′i, pb) =

χ(pa, ri, pb) if χ(pa, ri, pb) 6= 0

χ(pa, ri+1, pb) if χ(pa, ri, pb) = 0 and χ(pa, ri+1, pb) 6= 00 if χ(pa, ri, pb) = χ(pa, ri+1, pb) = 0

.

Second, for p in P and r′i in E, we have

χ′(r′i, r′i+1, p) =

1 if χ(ri, ri+1, p) 6= 0 or p = ri

−1 if χ(ri, ri+1, p) = 0 and p 6= ri.

Third, for p in P and r′i, r′j in E such that i is in 1, . . . , n − 1, j is in2, . . . , i− 1, we have

χ′(r′i, r′j, p) =

χ(ri, rj, p) if χ(ri, rj, p) 6= 0−χ(ri, ri+1, p) if χ(ri, rj, p) = 0 and p 6= rj

χ(ri, ri+1, p) if p = rj

.

r1r2

r3 r4

r1r2

r3 r4

r′1

r′2

r′3

r′4

FIGURE 6.10: An order type and his augmented order type.

We now have a proof for Theorem 6.2.2. Figure 6.11 gives an illustrationfor the second part of the proof.

86 Chapter 6. The realizability problem for convex geometries

Proof of Theorem 6.2.2. We reduce the problem from realizability of abstractorder type. Given an abstract order type (P, χ), we select an extreme pointand build its augmented order type (P′, χ′), which induces an AOT convexgeometry (P′, C). We show that (P′, C) is realizable if and only if (P, χ) isrealizable. We select an extreme point r1 of (P, χ) such that (r1, . . . , rk) is theextreme cycle.

First, if (P, χ) is realizable then there is a point set that has χ as its chiro-tope. Let r1 be an extreme point of the affine convex geometry induced bythis point set. Suppose that (r1, . . . , rk) is the extreme cycle of this point set.Now we add (r′1, . . . , r′k) in this order. Each r′i is placed close enough of ri torespect the orientations defined by the augmented order type with the pointin P and the points in r′1, . . . , r′i−1. We obtain a realization for (P′, χ′) and arealization of (P′, C). All the steps can be done in polynomial time.

Second, suppose that (P′, C) is realizable. By Lemma 6.2.7 the point set Qthat realizes the convex geometry has its extreme cycle equal to (r1, . . . , rk) or(rk, . . . , r1). So either Q or Q′, the image of Q by an axial symmetry, has itsextreme cycle equal to (r1, . . . , rk). Suppose Q has its extreme points orderedas (r1, . . . , rk). We call (P′, χ2) the (abstract) order type induced by Q. We useProposition 6.2.1 to conclude that χ′ = χ2. So (P′, χ′) is realizable (with thepoint set Q). We obtain a realization of (P, χ) by taking a sub-realization (i.e.a subset of point in the realization) of (P′, χ′). All the steps can be done inpolynomial time.

(P, χ) (P ∪ r′1, . . . , r′k, χ′) (P ∪ r′1, . . . , r′k, C)

Q(P ∪ r′1, . . . , r′k, χ2)

Abstract order type Augmented order type Induced convex geometry

RealizationOrder type induced by the point set

FIGURE 6.11: Second part of the proof for Theorem 6.2.2.

Proof of Theorem 6.2.1. We reduce from realizability of AOT convex geome-tries encoded by an abstract order type. So given an abstract order type(P, χ) that induces a convex geometry (P, C), we show that we can obtain,in polynomial time, the copoint poset of (P, χ).

Indeed, for such a convex geometry (P, C) (with |P| > 2) the copoints (i.e.the convex sets that are not the intersection of other convex sets) are among

6.2. Hardness result for the realizability problem 87

the sets of the form p ∈ P : χ(p1, p, p2) 6= 1 for p1, p2 in P. Note that aset p ∈ P : χ(p1, p, p2) 6= 1 does not necessarily give a copoint. So thecopoint poset of an AOT convex geometry has less than |P|2 elements andcan be quickly retrieved from the abstract order type.

This shows that the Edelman–Jamison problem is hard from a computa-tional point of view.

89

Chapter 7

Conclusion

“If you’re reading this, you’ve gotten out. And if you’ve come this far, maybeyou’re willing to come a little further.”

— Andy Dufresne in The Shawshank Redemption

We conclude this work by listing some open problems related to this the-sis and giving a closing remark about the work done.

7.1 Further work

Regarding the monophonically convex geometries, the algorithm that solvesthe maximum-weight convex set in chordal graphs (in Chapter 5) is mainlybased on the existence of clique separator graphs for chordal graphs. We be-lieve this structure is helpful to obtain a linear description of the convex setpolytope (defined in Chapter 4) based on monophonically convex geome-tries. That would solve a question asked by Korte and Lovasz [82]. But evenfor the monophonically convex geometries based on a split graph, this de-scription does not seem easy to obtain, even when considering the structuralresults obtained in Chapter 4. The investigation could start with the specialcase of k-trees, knowing that the case k = 1 is well known.

About the maximum-weight convex set problem for other convex geome-tries, there are several families for which we do not know whether there isa polynomial time algorithm that solves the maximum-weight convex setproblem. The technique used in this thesis relies on finding a link between aspecific family of convex geometries and the family of poset convex geome-tries. The set of AT-free convex geometries seems a good candidate to startwith. We believe that the structure that underlies the AT-free graphs can bemodeled by several posets.

90 Chapter 7. Conclusion

Another possible continuation is to adapt the maximum-weight convexset algorithm based on the clique separator graph for other structure builton chordal graphs, for instance, the clique elimination greedoid defined inGoecke et al. [54].

The hardness result in Chapter 6 could be helpful to show that other geo-metrical problems are ∃R-hard. The realization problem for convex geome-tries can also be extended to other families. For instance, given a copointposet of a convex geometry (V, C), is there a poset P such that (V, C) is theposet convex geometry induced by P? This decision problem is quite easy(solvable in polynomial time) because, by looking at the copoints (i.e. theideals induced by one element), we can try to reconstruct the cover relationsof a poset. If the construction leads to an actual poset, the answer is yes, oth-erwise, the answer is no. But the problem can be adapted for double posetconvex geometries or for monophonically convex geometries. It seems that alot of convex geometry families based on “strong” structures (e.g. graphs orposet) can be easily decided. However, further investigations are required.

7.2 Closing remark

This work provides some results on two natural problems about convex ge-ometries. First, the maximum-weight convex set problem. Studying thespecial case of the split graph leads us to a polynomial time algorithm forsolving the problem on monophonically convex geometries. We hope ourwork will pave the way for further research on monophonically convex ge-ometries. Second, the realizability problem. The hardness result for abstractorder types leads us to solve, from a computational point of view, a long-standing open question. We hope that more hardness results in computationgeometry will be derived from this result in further research.

91

Appendix A

Allowable sequences andrealizability problems

We provide here an idea of the original proof of the hardness result for therealizability problem of convex geometries. This original proof was obtainedin collaboration with Hoffmann [67] and relies on results for allowable se-quences that can be found in the Ph.D. thesis of Hoffmann [65].

A.1 Allowable sequences

In the following, we consider another combinatorial description of a pointset, the (generalized) allowable sequence or circular sequence. An allow-able sequence is a sequence of permutations π1, . . . , πk of the elements of1, . . . , n such that every pair of elements is reversed exactly once in the se-quence, and two consecutive permutations differ by a move, the reversal ofdisjoint (consecutive) substrings of the permutation. For instance, the follow-ing sequence Ω of permutations is an allowable sequence:

π1 = (1, 2, 3, 4, 5, 6)

π2 = (1, 3, 2, 4, 6, 5)

π3 = (1, 3, 6, 4, 2, 5)

π4 = (6, 3, 1, 4, 2, 5)

π5 = (6, 3, 4, 1, 2, 5)

π6 = (6, 3, 4, 5, 2, 1)

π7 = (6, 5, 4, 3, 2, 1)

π8 = (5, 6, 4, 2, 3, 1)

92 Appendix A. Allowable sequences and realizability problems

Each reversal in a move is called a switch, and a switch is simple if itinvolves a substring of size two. A special case of allowable sequences are thesimple allowable sequences, i.e. sequences where each move is one simpleswitch.

An allowable sequence can be obtained from a point set in R2 in the fol-lowing way. The orthogonal projection of the point set onto an oriented lineleads to a permutation of the points. Rotating this line (by a continuous mo-tion) by an angle of 180 leads to a sequence of permutations which is an al-lowable sequence. Figure A.1 gives an illustration of this procedure. A pointset in R2 in general position with the additional condition that no two linesspanned by different pairs of points are parallel induces a simple allowablesequences.

1 23

4

3 421

3214

3124

13241243

1423

4123

FIGURE A.1: An example of an allowable sequence of a pointset.

The fact that a point set induces an allowable sequence leads to the ques-tion of the realizability. An allowable sequence Π is realizable if there existsa point set P in the plane such that the allowable sequence induced by Pis Π. For instance, the sequence Ω given above is realizable. A point setthat realizes Ω is shown in Figure A.2. Note that realizable simple allowablesequences correspond to point sets in general position with the additionalcondition that no two lines spanned by different pairs of points are parallel.

1

2

3

5

4

6

FIGURE A.2: A point set that induces the sequence Ω.

A.1. Allowable sequences 93

The concept of an allowable sequence has first been described by Per-rin [102] in 1882 who conjectured that every allowable sequence is realizable.This was disproved by Goodman and Pollack [58] who showed that the “badpentagon”, illustrated by Figure A.3, is not realizable. This paper also in-troduces the term “allowable sequence” for point sets in general position.The authors generalized the definition in Goodman and Pollack [56] usingsequences of permutations. Allowable sequences have several applicationsin combinatorial geometry, see for example Goodman and Pollack [57]. Thecomputational complexity of the realizability problem has been posed as anopen question in Pilz [105].

π1 = (1, 2, 3, 4, 5)π2 = (2, 1, 3, 4, 5)π3 = (2, 1, 4, 3, 5)π4 = (2, 1, 4, 5, 3)π5 = (2, 4, 1, 5, 3)π6 = (4, 2, 1, 5, 3)π7 = (4, 2, 5, 1, 3)π8 = (4, 2, 5, 3, 1)

1

2

3

4

5

π9 = (4, 5, 2, 3, 1)π10 = (5, 4, 2, 3, 1)

FIGURE A.3: The non-realizable pentagon whose sides and“parallel” diagonals meet in a specific order.

One basic property describing the relation between allowable sequencesand order types is given by the following lemma due to Goodman and Pol-lack [58, 57].

Lemma A.1.1. An allowable sequence uniquely determines an abstract order type.

But a given abstract order type can be determined by several allowablesequences, as shown below. In other words, allowable sequences carry moreinformation than the abstract order type relation. Formally, given an allow-able sequence on P = 1, . . . , n we obtain an abstract order type (P, χ) bysetting, for p1, p2, p3 in P, χ(p1, p2, p3) = 0 if p1, p2, p3 are contained ina switch, χ(p1, p2, p3) = 1 if the switch involving p1 and p2 precedes theswitch involving p1 and p3, and χ(p1, p2, p3) = −1 otherwise. When we

94 Appendix A. Allowable sequences and realizability problems

mention this abstract order type connected to the allowable sequence we re-fer to it as the abstract order type induced by the allowable sequence. Fig-ure A.4 shows two distinct (realizable) allowable sequences that induce thesame (realizable) abstract order type.

1 4

2 3

π1 = (1, 2, 3, 4)

π2 = (2, 1, 3, 4)

π3 = (2, 3, 1, 4)

π4 = (3, 2, 4, 1)

π5 = (3, 4, 2, 1)

1 4

2 3

π1 = (1, 2, 4, 3)

π2 = (2, 1, 3, 4)

π3 = (2, 3, 1, 4)

π4 = (3, 2, 4, 1)

FIGURE A.4: Two allowable sequences inducing the same or-der type.

Similarly to abstract order types, we have the concept of realization spacefor a generalized allowable sequence A with n elements. If we identify natu-rally each point set on n elements with a point in Rn, the realization space ofA is the set of all points in R2n that give a realization of A in R2.

A.2 Realizability for allowable sequence

Using ideas form Shor’s proof [115] of Mnev’s Universality Theorem [93],we show that the same type of universality as for order types also holds forallowable sequences. This leads to the following result.

Theorem A.2.1. The realizability problem for generalized allowable sequences is∃R-complete.

The main idea is to show an “equivalence” between the primary semi-algebraic sets (i.e. sets in Rn defined by a polynomial inequality systemsthat consists of strict inequalities and equalities) and the realization spacesfor a generalized allowable sequences. The notion of “stable equivalent” isinvolved and would require further details (but it implies “homotopy equiv-alence”). For a more formal definition, see Richter-Gebert [110]. The proof ofTheorem A.2.1 uses the following theorem due to Shor [115].

Theorem A.2.2. Every primary semi-algebraic set V ⊆ Rd is stably equivalent toa semi-algebraic set V′ ⊆ Rn, with n = poly(d), for which all defining equationshave the form xi + xj = xk or xi · xj = xk for certain 1 ≤ i ≤ j < k ≤ n, where thesequence 1 = x1, x2, . . . , xn is always strictly increasing.

A.3. Results for simple allowable sequences 95

In addition to the decomposition of a polynomial equation into singleaddition or multiplication steps, the theorem also gives us a total order ofthe variables. We call the description of the semi-algebraic set as in Theo-rem A.2.2 one of its normal form.

The proof of the equivalence also relies on the classic construction of vonStaudt [118] involving the cross-ratio. The idea of von Staudt constructions isto encode an arbitrary system of polynomial equations and strict in equalitiesinto the geometry of a point configuration, as shown in Figure A.5. The cross-ratio (a, b; c, d) of four points a, b, c, d ∈ R2 on a line is defined as

(a, b; c, d) :=|a, c| · |b, d||a, d| · |b, c| ,

where |x, y| is the oriented euclidean distance between x and y. The cross-ratio is invariant under projective transformations, so we can extend the def-inition to the case when (exactly) one of the point is at infinity: the two dis-tances involving that point are dropped from the formula. If all of a, b, c andd are infinite points with slopes a′, b′, c′, and d′ then their cross-ratio (a, b; c, d)is

(c− a)(d− a)

(b− d)(b− c)

.

10 x y xy0 x y x+ y ∞

`∞ `∞

∞

B B

ai

cidi

aibi

ci

di

bi

` `

FIGURE A.5: Gadgets for addition (left) and multiplication(right) on a line.

A.3 Results for simple allowable sequences

It is possible to extend Theorem A.2.1 to simple allowable sequences, whichis a useful tool to show the ∃R-hardness of other geometric realizability prob-lems. Therefore we need to construct a simple allowable sequence Ω′ from ageneralized allowable sequence Ω such that Ω′ is realizable if and only if Ω

96 Appendix A. Allowable sequences and realizability problems

is realizable. We describe the construction geometrically from a realization.A similar combinatorial allowable sequence can be constructed for any givennon-realizable allowable sequence.

Theorem A.3.1. The realizability problem of simple allowable sequences is ∃R-complete.

To prove the theorem above we extend the notion of constructible ordertypes, that has already been used to show that simple order type realizabilityis ∃R-complete, see Bjorner et al. [15], Shor [115] and Matousek [91]. Themain idea consist of replacing the points that prevent the allowable sequencefrom being simple. Every point, that lies on lines spanned by other points,is replaced by four new points as shown in Figure A.6. Next, the pointsthat induce parallel lines can be slightly moved to obtain a simple allowablesequence.

`1

`2

p

p4 p1

p2p3

FIGURE A.6: Replacing a point p by four points.

A.4 Realizability for convex geometries

We show that the realizability of AOT convex geometries, encoded by anabstract order type, is ∃R-hard. We proceed to define some abstract ordertypes which are the building blocks for our reduction (from realizability ofsimple allowable sequences). We define (D2k, χ2k) as the (abstract) order typeof the following point set in the plane. Consider k lines that intersect in thecommon center point of two circles of almost the same radius. Place a pointon each intersection point of any of the two circles with any of the k lines.We denote the points on the outer circle by ri and the inner ones by r′i in

A.4. Realizability for convex geometries 97

counterclockwise order. The difference of the radii is small enough, such thatr′i is an extreme point of D2k \ ri and the only collinearities in D2k appearamong the points ri, r′i, ri+k and r′i+k.

lies on the convex hull (of the set) when we remove ri. We call D2k a dou-ble ring and denote the induced (AOT) convex geometry by (D2k, C2k). Forri or r′i, we assume for convenience that an index i larger than 2k is identifiedwith an element of 1, . . . , k as follow. The index is identified with 2k if i is amultiple of 2k and with i mod 2k otherwise.

Because the only collinearities in D2k appear among the points ri, r′i, ri+k

and r′i+k. This means we can slightly perturb the lines in the constructionof D2k, such that they do not intersect at the center but form an arbitraryline arrangement in the neighborhood of the center, and still obtain the sameabstract order type, as illustrated by Figure A.7.

r1 r′1

r2 r′2

r3 r′3 r′4 r4

r′5r5

r6r′6

FIGURE A.7: Small perturbations in the double ring.

We will use this perturbation and a “unique representation” we obtainfrom Lemma A.4.1, which we will prove in the rest of this subsection, tofix an allowable sequence of a point set with a double ring. We have thefollowing lemma, which has the same purpose that Lemma 6.2.7.

Lemma A.4.1. In each realization of (D2k, C2k) the order of the extreme points is(up to a reflection) (r1, . . . , r2k).

To show Theorem 6.2.2 using allowable sequences, we set up a reductionfrom realizability of simple allowable sequences. We give here only the mainidea of the proof and omit some technical verification (for a formal proofof the theorem, see Chapter 6). Figure A.8 shows the objects involved inthe reduction below. Given a simple allowable sequence A (with ground setP = 1, . . . , n), we build an abstract order type OA which induces an AOTconvex geometry. The abstract order type OA will be called modified ordertype of A, we describe its construction as follows. Let PA be the abstract

98 Appendix A. Allowable sequences and realizability problems

order type that is induced by the allowable sequence A (see Chapter 6). Todefine OA we add a double ring D2k to PA, where k is the number of switchesin the allowable sequence (k = (n

2) because A is simple). This will be donein such a way that the double ring forms the two ”outer layers” of the pointsin PA. Formally, to complete the definition of OA, we have to define theorientation of triples containing two points of PA and one point of D2k andvice versa. First, the points of PA that are reversed in the i-th switch of Aare collinear with ri, ri+k and r′i, r′i+k of D2k. Second, we have to define theorientation of non-collinear triples. A triple (ph, pi, rj) with j ∈ 1, . . . , k− 1is oriented clockwise if ph appears before pj in the j-th permutation of A, andcounterclockwise otherwise. The orientation of (ph, pi, rj+k) is the reverse of(ph, pi, rj). A triple (rh, ri, pj) is oriented clockwise if i ∈ h+ 1, . . . , h+ k− 1and counterclockwise if i ∈ h− 1, . . . , h− (k− 1). The triple including r′hhas the same orientation as the triple where r′h is replaced by rh. If rh and r′hare included in a triple then, the latter has the same orientation as the triplewhere r′h is replaced by rh+k.

A = (P,Π) PA = (P, χ) OA = (P ∪D2k, χ′)

CA = (P ∪D2k, C)

Simple allowable seq. Induced abstract order type Modified order type

Induced convex geometry

FIGURE A.8: Relations between A, PA, OA and CA.

The reduction from realizability of simple allowable sequences is similarto the proof of Theorem 6.2.2. To establish the reduction let as in the previousparagraph A be an allowable sequence on P, OA its modified order type andCA the convex geometry induces by OA.

Suppose that CA is realizable, we know that each realization of CA (en-coded by OA) contains a copy of the double ring (i.e. the points in D2k), withthe same order of the extreme points. With an argument similar to the oneused in the proof of Proposition 6.2.1, we show that OA is also realizable. Ina realization of OA, we have that the line spanned by rk and rk+n contains thepoints of P that are reversed in switch k in the allowable sequence. In otherwords, the double ring fixes the order of slopes of the lines spanned by twopoints in P. We obtain the allowable sequence P by considering the allowablesequence induced by a sub-realization of OA. So P is also realizable.

A.4. Realizability for convex geometries 99

Now, suppose that A is realizable. We can realize OA by placing the pointsof the double ring D2k on the intersection points of the lines spanned by Pwith two “very large” circles of almost the same radius that contain a real-ization of A “close” to their center point.

101

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113

Index

abstract order type, 73accessibility property, 12acyclic oriented matroid of rank 3, 7alignment, 10allowable sequence, 71

generalized, 91simple, 92

anti-exchange property, 10antimatroid, 12

(a, b)-path shelling, 25chordal graph, 22line-search, 24point-search, 24double poset, 20edge shelling of a tree, 21line-search

undirected, 24point-line search, 24point-search

undirected, 24poset, 19, 42split graph shelling, 36, 42vertex shelling of a tree, 21

arc, 6asteroidal triple, 25attached at, 14augmented order type, 84

base, 15, 44poset, 15

basic word, 14

114 INDEX

blockingK-blocking, 58

c-convex, see convex set, connectedcharacteristic vector, 31chirotope, 73

acyclic rank 3 chirotope, 73circuit, 15circular sequence, 91clique, 6, 36

maximal, 6clique nodes, 52clique-separator graph, 52closed set, 10closure operator, 10complexity class, 7connected, 6connected component, 6convex geometry, 10

AOT, 23AT-free, 25connected, 21geodesic, 21lower affine, 22monophonically, 22transitive, 24uncover, 25affine, 22, 36double poset, 20, 33on edges of a tree, 21, 35on vertices of a tree, 21, 34poset, 19, 31

convex hull, 7convex set, 10, 13, 51

geodesic, 21connected, 21monophonically, 21

INDEX 115

polytope, 31, 33, 35transitive, 23

copoint, 14poset, 15

cover, 6cross-ratio, 95cycle, 6

chordless, 6

decomposable, 35monotone, 35

double ring, 97downset, see idealdummy vertex, 54

edge, 5elminiation process, 13endpoint, 15Euclidean convex set, 7exchange property, 11existential theory of the reals, 76extended, 54extension, 54extract, 51extreme cycle, 79extreme point, 12, 78

factor α approximation, 7feasible continuation, 16feasible set, 12, 13∗-feasible, 40i-feasible, 40polytope, 31

filter, 6fixed ring, 83forced set function, 41free set, 15

g-convex, see convex set, geodesic

116 INDEX

graph, 5s, t-graph, 6AT-free, 25block, 21chordal, 6directed, 5Ptolemaic, 21split, 21, 36

greedoid, 17ground set, 5

Hasse diagram, 7

immediate successor, 34indeal, 6independent set, 6, 36induce, 94interior, 16isomorphic, 74

leaf, 21

m-convex, see convex set, monophonicallymodified order type, 97monophonically convex set, 51move, 91

normal form, 95

order type, 73abstract, 7, 23

partially ordered set, 6path, 6

chordless, 6poset, see partially ordered set

K-rooted, 56predecessor, 34primary semi-algebraic set, 94problem

INDEX 117

maximum closure, 46maximum-weight

convex set, 28, 53feasible set, 28, 46

pseudoline, 75

realizability, 72realizability problem, 74

for convex geometry, 72realizable, 74, 92realization space, 94representation, 75root

of a circuit, 16rooted

R-rooted, 54subset, 16

separator, 51uv-separator, 52minimal, 51

separator nodes, 52set system, 5shelling, 13shelling process, 13simplicial, 6sink, 6source, 6stem, 16subgraph, 6subtree, 34, 53switch, 92

simple, 92

t-convex, see convex set, transitivetrace, 15tree, 20

k-tree, 59

118 INDEX

unforced set function, 41universal extreme, 25upset, see filter

vertex, 5vertex separator

minimal, 52

weight function, 5