A New Facet Generating Procedure for the Stable Set Polytope › presentation › 0c8a › 1021f26fa06… · Graph coloring (Cornaz & Jost (2008)) 4 / 22. Stable Set Polytope: Applications

A New Facet Generating Procedurefor the Stable Set Polytope

Alinson S. Xaviera Manoel Campelob

a Mestrado e Doutorado em Ciencia da ComputacaoUniversidade Federal do Ceara

Fortaleza, Brazil

b Departamento de Estatıstica e Matematica AplicadaUniversidade Federal do Ceara

Fortaleza, Brazil

LAGOS’11

1 / 22

Outline

1. Introduction

2. Preliminaries

3. The Procedure

4. Example

5. Concluding Remarks

2 / 22

Stable Set Polytope: Definition

� Given a simple, undirected graph G = (V ,E)

Definition

A stable set is a set of pairwise non-adjacent vertices.

Example Not an example

Definition

The stable set polytope is the convex hull of the incidence vectors of all thestable sets:

STAB(G) = conv{x ∈ {0, 1}V : xu + xv ≤ 1, ∀{u, v} ∈ E}

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Stable Set Polytope: Definition

� Given a simple, undirected graph G = (V ,E)

Definition

A stable set is a set of pairwise non-adjacent vertices.


Definition

The stable set polytope is the convex hull of the incidence vectors of all thestable sets:

STAB(G) = conv{x ∈ {0, 1}V : xu + xv ≤ 1, ∀{u, v} ∈ E}

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Stable Set Polytope: Applications

� Applications in various fields (Bomze, Budinich, Pardalos & Pedillo (1999)):

� Coding theory

� Computer vision and pattern matching

� Molecular biology

� Scheduling

� Model other important combinatorial problems:

� Set packing and set partitioning (Padberg (1973))

� Graph coloring (Cornaz & Jost (2008))

4 / 22

Stable Set Polytope: Applications

� Applications in various fields (Bomze, Budinich, Pardalos & Pedillo (1999)):

� Coding theory

� Computer vision and pattern matching

� Molecular biology

� Scheduling

� Model other important combinatorial problems:

� Set packing and set partitioning (Padberg (1973))

� Graph coloring (Cornaz & Jost (2008))

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Previous Research: Facet Producing Subgraphs

� Facet producing subgraphs:

� Cliques (Padberg (1973))

� Odd holes (Padberg (1973))

� Odd anti-holes (Nemhauser & Trotter (1974))

� Webs and anti-webs (Trotter (1975))

� Subdivided wheels (Cheng & Cunningham (1996))

� others...

Example

1

2

34

5

P5i=1 xi ≤ 1

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� others...

Example

12

3

45

6

7

P7i=1 xi ≤ 3

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� others...

Example

12

3

45

6

7

P7i=1 xi ≤ 2

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� others...

Example

12

3

45

6

7

8

3x8 +P7

i=1 xi ≤ 3

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� others...

Example

12

3

45

6

7

8

3x8 +P7

i=1 xi ≤ 3

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Previous Research: Facet Generating Procedures

� Facet generating procedures:

� Subdividing edges (Wolsey (1976))

� Subdividing stars (Barahona & Mahjoub (1994))

� Replacing vertices with stars (Canovas, Landete & Marın (2003))

� Replacing edges with gears (Galluccio, Gentile & Ventura (2008))

� others...

Example

1

2

34

56

1

2

34

5

6

8

7

2x6 +P5

i=1 xi ≤ 2 2x6 +P5

i=1 xi + x7 + x8 ≤ 3

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� others...

Example

1

2

34

56

1

2

34

56

78

910

11

2x6 +P5

i=1 xi ≤ 2 3x6 +P5

i=1 xi +P11

i=7 xi ≤ 5

6 / 22







� others...

Example

1

6 7 8

4 3

5 2

1

4 3

5 2

x1 +P5

i=2 xi ≤ 1 2x1 +P5

i=2 xi +P8

i=6 xi ≤ 3

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� others...

Example

7

2

1 3 1 3

6

11

8

14 5

2

10

5 4

P3i=1 xi + x4 + x5 ≤ 2

P3i=1 xi + 2x4 + 2x5 +

P11i=6 xi ≤ 4

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� others...

Example

7

2

1 3 1 3

6

11

8

14 5

2

10

5 4

P3i=1 xi + x4 + x5 ≤ 2

P3i=1 xi + 2x4 + 2x5 +

P11i=6 xi ≤ 4

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The New Procedure

� Replace (k − 1)-cliques with certain k-partite graphs

� Particular case: replace vertices with bipartite graphs

� Unifies and generalizes many previous procedures:

� Subdividing edges

� Subdividing stars

� Replacing vertices with stars

� Generates many new facet-defining inequalities

� The idea that leads to the procedure can be used with other polytopes

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The New Procedure









7 / 22

The New Procedure









7 / 22

The New Procedure









7 / 22

Preliminaries

8 / 22

Affine Isomorphism

Definition (Ziegler (1994))

Two polytopes are affinely isomorphic (not. P1∼= P2) if there is an affine map

that is a bijection between their points.

� Implies combinatorial equivalence

� Easy to transform facets of P1 into facets of P2

9 / 22

Affine Isomorphism






9 / 22

Affine Isomorphism






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Facets from Faces: Overview

� Convert facets of a face F into facets of the polytope P:

� Find a sequence F1, . . . ,Fk such that:

� F1 = F and Fk = P

� Fi is a facet of Fi+1

� Repeatedly apply the following theorem:

Theorem

Let P be a convex polytope and S a finite set such that P = conv(S). If cx ≤ dis facet-defining for P and πx ≤ π∗ is facet-defining for {x ∈ P : cx = d}, thenπx + α(cx − d) ≤ π∗ is facet-defining for P, where

α = max

π∗ − πx

cx − d: x ∈ S , cx < d

ff

� Extension of the sequential lifting procedure

� Different sequences may yield different facets

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Theorem


α = max

π∗ − πx

cx − d: x ∈ S , cx < d

ff



10 / 22







Theorem


α = max

π∗ − πx

cx − d: x ∈ S , cx < d

ff



10 / 22







Theorem


α = max

π∗ − πx

cx − d: x ∈ S , cx < d

ff



10 / 22

Facets from Faces: Geometric View

Example

� P = conv{(0, 0), (2, 0), (2, 1), (1, 2), (0, 1)}

� F = {x ∈ P : x1 + x2 = 3}� −x1 ≤ −1 is facet-defining for F

� α = maxn−x1+1

3−x1−x2: x ∈ P, x1 + x2 < 3

o= 1

2

� −x1 + 12(x1 + x2 − 3) ≤ −1

� −x1 + x2 ≤ 1 is facet-defining for P

11 / 22


Example

� P = conv{(0, 0), (2, 0), (2, 1), (1, 2), (0, 1)}� F = {x ∈ P : x1 + x2 = 3}

� −x1 ≤ −1 is facet-defining for F


3−x1−x2: x ∈ P, x1 + x2 < 3

o= 1

2

� −x1 + 12(x1 + x2 − 3) ≤ −1


11 / 22


Example

� P = conv{(0, 0), (2, 0), (2, 1), (1, 2), (0, 1)}� F = {x ∈ P : x1 + x2 = 3}� −x1 ≤ −1 is facet-defining for F


3−x1−x2: x ∈ P, x1 + x2 < 3

o= 1

2

� −x1 + 12(x1 + x2 − 3) ≤ −1


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Example



3−x1−x2: x ∈ P, x1 + x2 < 3

o= 1

2

� −x1 + 12(x1 + x2 − 3) ≤ −1


11 / 22


Example



3−x1−x2: x ∈ P, x1 + x2 < 3

o= 1

2

� −x1 + 12(x1 + x2 − 3) ≤ −1


11 / 22


Example



3−x1−x2: x ∈ P, x1 + x2 < 3

o= 1

2

� −x1 + 12(x1 + x2 − 3) ≤ −1


11 / 22

Strong Hypertrees: Definitions

Definition

Two hyperedges are strongly adjacent if they have same size k and share exactlyk − 1 vertices.


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Definition

A hypergraph H = (V , E) is a strong hypertree either if E = {V } or there is aleaf v incident to an hyperedge e such that:

� e is strongly adjacent to some other hyperedge,

� (V \ {v}, E \ {e}) is also a strong hypertree.

Example

(hyperedges are represented as cliques)

Example

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Definition




Example


Example

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Definition




Example


Example

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Definition




Example


Example

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Definition




Example


Example

12 / 22


Definition




Example


Example

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Definition




Example


Example

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Definition




Example


Example

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Definition




Example


Example

12 / 22


Definition




Example


Example

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Definition




Example


Example

12 / 22


Definition




Example


Example

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Definition




Not an exampleNot an example

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Definition

A strong hypertree is a strong hyperpath if it has exactly two leaves.

Example

12 / 22

Strong Hypertrees: Properties

� Similarly to ordinary trees:

Proposition

Every strong hypertree with n vertices:

� is a k-uniform hypergraph, for some k

� has n − k + 1 hyperedges

� has a full rank incidence matrix

� contains a strong hyperpath between any pair of non-adjacent vertices

Example

k = 3 |V | = 10 |E| = 8

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Proposition






Example

k = 3 |V | = 10 |E| = 8

13 / 22



Proposition






Example

k = 3 |V | = 10 |E| = 8

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The Procedure

14 / 22

The Procedure: Context

� Q: set of maximal cliques of G

� C(G) = (V ,Q): clique hypergraph of G

� T = (VT ,QT ) ⊆ C(G): A k-uniform strong hypertree such that:

� G [VT ] is k-partite with classes V1, . . . ,Vk

� No vertex in V0 := V \ VT has neighbors in all classes V1, . . . ,Vk

� Consider the face FT = {x ∈ STAB(G) : xQ = 1,∀Q ∈ QT}

Example

7

6 8

1

2 53 4

V0 = {6, 7, 8}V1 = {2, 4}V2 = {3, 5}V3 = {1}

FT =

8<:x ∈ STAB(G) :x1 + x2 + x3 = 1x1 + x3 + x4 = 1x1 + x4 + x5 = 1

9=;

15 / 22

The Procedure: Context

� Q: set of maximal cliques of G

� C(G) = (V ,Q): clique hypergraph of G

� T = (VT ,QT ) ⊆ C(G): A k-uniform strong hypertree such that:

� G [VT ] is k-partite with classes V1, . . . ,Vk

� No vertex in V0 := V \ VT has neighbors in all classes V1, . . . ,Vk

� Consider the face FT = {x ∈ STAB(G) : xQ = 1, ∀Q ∈ QT}

Example

7

6 8

1

2 53 4

V0 = {6, 7, 8}V1 = {2, 4}V2 = {3, 5}V3 = {1}

FT =

8<:x ∈ STAB(G) :x1 + x2 + x3 = 1x1 + x3 + x4 = 1x1 + x4 + x5 = 1

9=;15 / 22

The Procedure: Lemmas 1 and 2

Lemma

The dimension of FT is |V | − |QT |.

� Each hyperedge we remove from T increases the dimension of FT byexactly one (if the remaining hypergraph is still a strong hypertree)

� Useful when building the sequence of faces

Lemma

If x ∈ FT , then xu = xv , for all u, v ∈ Vi , i ∈ {1, . . . , k}.

� Each class can be seen as a single vertex

� Vk is selected if and only if no other class is selected

16 / 22


Lemma




Lemma




16 / 22


Lemma




Lemma




16 / 22


Lemma




Lemma




16 / 22

The Procedure: Lemma 3

� Consider the graph GT obtained from G by:

� Contracting each class Vi into a vertex vi ∈ Vi , for i ∈ {1, . . . , k − 1}� Removing the vertices of Vk

Lemma

If V0 has no neighbors in VK then FT is affinely isomorphic to STAB(GT ).

Example

7

6 8

1

2 53 4

6

7

8

5 4

17 / 22




Lemma


Example

17 / 22




Lemma


Example

17 / 22




Lemma


Example

17 / 22




Lemma


Example

17 / 22




Lemma


Example

17 / 22




Lemma


Example

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The Procedure: Main Theorem

Theorem

Suppose that V0 has no neighbors in VK . IfXv∈V (GT )

πv xv ≤ π∗ (1)

is facet-defining for STAB(GT ), then there exists αQ , ∀Q ∈ QT , such thatXv∈V (GT )

πv xv +X

Q∈QT

αQ(xQ − 1) ≤ π∗ (2)

is facet-defining for STAB(G).

Sketch of the proof.

� The inequality (1) is facet-defining for FT (lemma 3)

� Remove one hyperedge at a time from T , but keeping it a strong hypertree

� Each removal increases dim(FT ) by exactly one (lemma 1)

� We have the sequence needed for the procedure outlined previously

18 / 22

Example

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The Procedure: Example 1

Example

1

2 5

7

6 83 4

�P8

i=4 xi ≤ 1 is facet-defining for STAB(GT )

� α = max

8<:P8i=4 xi − 1 : x ∈ STAB(G),

x1 + x2 + x3 = 0,x1 + x3 + x4 = 1,x1 + x4 + x5 = 1

9=; = 1

20 / 22


Example

7

6 8

5 4

�P8

i=4 xi ≤ 1 is facet-defining for STAB(GT )

� α = max

8<:P8i=4 xi − 1 : x ∈ STAB(G),

x1 + x2 + x3 = 0,x1 + x3 + x4 = 1,x1 + x4 + x5 = 1

9=; = 1

20 / 22


Example

1

2 5

7

6 83 4

�P8

i=4 xi ≤ 1 is facet-defining for FT

� α = max

8<:P8i=4 xi − 1 : x ∈ STAB(G),

x1 + x2 + x3 = 0,x1 + x3 + x4 = 1,x1 + x4 + x5 = 1

9=; = 1

20 / 22


Example

1

2 5

7

6 83 4

�P8


� α = max

8<:P8i=4 xi − 1 : x ∈ STAB(G),

x1 + x2 + x3 = 0,x1 + x3 + x4 = 1,x1 + x4 + x5 = 1

9=; = 1

20 / 22


Example

2

7

6 8

53 4

1

�P3

i=1 xi +P8


� α = max

8<:P3i=1 xi +

P8i=4 xi − 2 : x ∈ STAB(G),

x1 + x2 + x3 = 0,

x1 + x3 + x4 = 0,x1 + x4 + x5 = 1

9=; = 1

20 / 22


Example

2 5

6 83 4

7

1

�P3

i=1 xi +P8


� α = max

8<:P3i=1 xi +

P8i=4 xi − 2 : x ∈ STAB(G),

x1 + x2 + x3 = 0,

x1 + x3 + x4 = 0,x1 + x4 + x5 = 1

9=; = 1

20 / 22


Example

2

7

6 8

53 4

1

� 2x1 + x2 + 2x3 + 2x4 +P8


� α = max

8<:2x1 + x2 + 2x3 + 2x4 +P8

i=5 xi − 3 : x ∈ STAB(G),

x1 + x2 + x3 = 0,x1 + x3 + x4 = 0,

x1 + x4 + x5 = 0

9=; = 0

20 / 22


Example

2 5

7

6 83 4

1

� 2x1 + x2 + 2x3 + 2x4 +P8


� α = max

8<:2x1 + x2 + 2x3 + 2x4 +P8

i=5 xi − 3 : x ∈ STAB(G),

x1 + x2 + x3 = 0,x1 + x3 + x4 = 0,

x1 + x4 + x5 = 0

9=; = 0

20 / 22


Example

2

7

6 8

53 4

1

� 2x1 + x2 + 2x3 + 2x4 +P8

i=5 xi ≤ 3 is facet-defining for STAB(G)

� α = max

8<:x4 + x5 + x6 + x7 + x8 − 1 : x ∈ STAB(G),x1 + x2 + x3 = 0,x1 + x3 + x4 = 1,x1 + x4 + x5 = 1

9=; = 1

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Concluding Remarks

� We presented a new facet generating procedure which:

� Unifies and generalizes many previous procedures

� Generates new facet-defining inequalities

� We also introduced the concept of strong hypertree

� Future research:

� Use the procedure to find new facet producing subgraphs

� Apply the main idea to other polytopes:

� Find faces that are affinely isomorphic to known polytopes

� Convert their facets into facets of the polytope

21 / 22

Concluding Remarks










21 / 22

Concluding Remarks










21 / 22

Thank you!

22 / 22

Documents

A New Facet Generating Procedure for the Stable Set Polytope › presentation › 0c8a › 1021f26fa06… · Graph coloring (Cornaz & Jost (2008)) 4 / 22. Stable Set Polytope: Applications