The Rectilinear Symmetric Crossing Minimization Problem

Seokhee Hong

The University of Sydney

1. Introduction

Graph drawing algorithm aims to construct geometric representations of graphs in 2D and 3D.

A - B, C, DB - A, C, DC - A, B, D, ED - A, B, C, EE - C, D

The input is a graph with no geometry

A B

D

C

E

The output is a drawing of the graph; the drawing should be easy to understand, easy to remember, beautiful.

Aesthetics in Graph Drawing

NP-hardness

• minimize edge crossings

• minimize area

• maximize symmetry

• minimize total edge length

• minimize number of bends

• angular resolution

• aspect ratio

• ……

Conflict

Minimize edge crossings

Maximize symmetry

• To construct symmetric drawings of graphs,

we need to solve two problems.

problem 1. find symmetry in graphs.

• symmetry finding algorithm

problem 2. display the symmetry in a drawing.

• symmetric drawing algorithm

• This talk: we want to construct a straight-line drawing which displays given symmetries with minimum number of edge crossings.

Symmetric Graph Drawing

Types of Symmetric Drawing in 2D

Types of symmetry in 2D

1. Rotation

2. Reflection

Types of symmetric drawing in 2D

1. Cyclic group of size k: k rotations

2. Dihedral group of size 2k: k rotations and k reflections

3. One reflection

Example: The Peterson Graph

dihedral group(size 10)• 5 rotations• 5 reflections

5 edge crossings

dihedral group(size 6)• 3 rotations• 3 reflections

3 edge crossings

1 reflection2 crossings

The Crossing Minimization Problem

A great deal of literature• Mathematicians: Crossing number [Pach]• Graph Drawers : Two layer crossing minimization [the book]

• [Buchheim & Hong 02]: crossing minimization for given symmetries with curve

The Symmetric Crossing Minimization Problem

8

• An automorphism of a graph G is a permutation of the vertex set which preserves adjacency.– Automorphism problem: isomorphism-complete

2. Background

• [Eades & Lin 00]

geometric automorphism is an automorphism which can be displayed as a symmetry of a drawing D of G.– Geometric automorphism problem: NP-complete

1

5

43

2 1 2 3

4 5

(12)(5)(34) (123)(45)

9

Geometric Automorphism Group

[Eades & Lin 00]

a group of geometric automorphisms which can be displayed as symmetries of a single drawing D of G.

1

2 3

41 2

3 4

(1234)(123)(4)

Symmetry finding step: find geometric automorphism group of G with maximum size

Symmetry Finding Algorithm

General graph• NP-hard

• Heuristics

• Exact algorithm: ILP• Exact algorithm: MAGMA

[Manning 90]

[Lipton,North,Sandberg85]

[de Fraysseix 99]

[Buchheim, Junger 01]

[Abelson,Hong,Taylor02]

[Manning, Atallah 88]

[Manning, Atallah 92]

[Manning 90]

[Hong, Eades, Lee 98]

[Hong, Eades 01]

Planar graph• Tree• Outerplanar graph• Plane graph• Series parallel digraph• Planar graph• triconnected planar graph

2D 3D

[Hong, Eades 00]

[Hong, Eades 00]

[Hong01]

[Hong01]

[Abelson, Hong,

Taylor 02]

N/A

N/A

[Hong, McKay,Eades02]

• Step1: symmetry finding algorithm– orbit: partition of vertex set V under the geometric

automorphism group– For each vertex v, the orbit of v under the group H is

the set { u V: u = p(v) for some p H}.• Step 2: drawing algorithm: [Eades & Lin 00]

– draw each orbit in a concentric circle• different drawings depending on the order of orbits

Output of [Abelson, Hong, Taylor 02]

8

1

2

3

4

5

6

7 9O1: {1}O2: {2,3,4,5}O3: {6,7,8,9}

Dodecahedron: dihedral 2D

4-cube: cyclic 2D

Fourcell: dihedral 2D|V| = 120, |E| = 1440

PSP44: cyclic 2D|V|=85, |E|=1700

[Buchheim and Hong02]

Symmetric Crossing Minimization Problem (SCM)

Input: A graph G, and a geometric automorphism p.

Output: A drawing of G which displays p with the minimum number of edge crossings.

[Theorem] The problem SCM is NP-hard.

Symmetric Crossing Minimization Problem

3. Rectilinear Symmetric Crossing Minimization

Rectilinear Symmetric Crossing Minimization Problem: (REC-SCM)

Input: A graph G, and a geometric automorphism group H.

Output: A straight-line drawing of G which displays H with the minimum number of edge crossings.

[Theorem] The problem REC-SCM is NP-hard.

<proof> We divide into three cases.

Cyclic Rectilinear Symmetric Crossing Minimization Problem: REC-SCM-Cyclic

Input: A graph G and a cyclic geometric automorphism group C.

Output: A straight-line drawing of G which displays C with the minimum number of edge crossings.

Dihedral Rectilinear Symmetric Crossing Minimization Problem: REC-SCM-Dihedral

Input: A graph G and a dihedral geometric automorphism group D.

Output: A straight-line drawing of G which displays D with the minimum number of edge crossings.

Axial Rectilinear Symmetric Crossing Minimization Problem: REC-SCM-Axial

Input: A graph G and an axial geometric automorphism group A.

Output: A straight-line drawing of G which displays A with the minimum number of edge crossings.

[Theorem] The problems REC-SCM-Cyclic, REC-SCM-Dihedral and REC-SCM-Axial are NP-hard.

<proof>• reduce the rectilinear crossing number problem, which

is NP-hard[Bienstock93] to REC-SCM problem.• use the similar argument from [Buchheim & Hong02].• main idea: define a function using disconnected

isomorphic graphs.

G

SCM : REC-SCM

• symmetric crossing number SCR(G,H): the smallest number of the edge crossings of a drawing of the graph G which displays the geometric automorphism group H.

• rectilinear symmetric crossing number REC-SCR(G,H): the smallest number of the edge crossings of a straight line drawing of the graph G which displays the geometric automorphism group H.

• [Lemma] SCR(G,H) REC-SCR(G,H)

Example: SCR(K6,H) REC-SCR(K6,H)

Two symmetric drawings of K6 displaying a dihedral group H of size 12.

SCR(K6, H) = 9 REC-SCR(K6,H) = 15

The Orbit Graphs• intra-orbit edge: if e connects two vertices in the same orbit. • inter-orbit edge: if e connects two vertices in two different

orbits.• The orbit graph G_H of G under H: a graph whose vertices

represent each orbit and edges represent the set of inter-orbit edges.

1

2

3

4

5

6

7 9

8

6,7,8,9

2,3,4,5

1

Choosing a Representation

• consider a graph with a single orbit.• different drawings displaying a given geometric

automorphism group H, depending on the choice of the representation.

• example: H = <p> = <(12345)>

1

2

3

45

1

2 3

4 5

1

234

5

1

2

3 4

5

p p2 p3 p4

More than Two Orbits• different drawings displaying a given geometric

automorphism group H, depending on the ordering of orbits.• example: two drawings displaying dihedral group of size 10

5 crossings 10 crossings

Heuristics for REC-SCM

• the main task: to decide the ordering of orbits to minimize edge crossings.

• The REC-SCM problem is geometric, rather than combinatorial, in contrast to the SCM problem.

10 crossings 15 crossings

Example: the radius makes a difference for REC-SCM.

4. Work in Progress

• SCM problem: evaluation of the general framework for SCM of [Buchheim and Hong 02]

• REC-SCM problem: – design and implement good heuristics (approximation

algorithms) to compute the ordering of the orbit graph. – variations of heuristics for the minimum linear

arrangement problem– variations of two layer crossing minimization methods– soft computing method: simulated annealing

Preliminary results

• Preliminary results showed that for the REC-SCM problem, simple heuristics perform fairly well for reasonably-sized graphs. – the size of the orbit graph is relatively small

compare to the size of the graph G.

• Observations:– there is a trade off between the quality and the run

time in general. – For dense graphs, the resulting drawings do not

appear different to human eyes. – For sparse graphs, it is easier to compare the

result instantly.

Future Work

• So far, we mainly focused on displaying the cyclic group.

• Further work includes the axial case and the dihedral

case.

• However, this looks challenging, because of the existence of edges that are fixed by the axial symmetry.

5. Open Problems• Find a tight lower bound of the SCR and REC-SCR:

– Intuitively, a drawing that displays a larger geometric automorphism group needs more crossings.

• Investigate the relationship between the crossing numbers such as CR, SCR, REC-CR, REC-SCR:

– CR(G) REC-CR(G)

– SCR(G,H) REC-SCR(G,H)

– CR(G) SCR(G,H) : CR(G) = SCR(G,H), where H is a trivial automorphism group

– REC-CR(G) REC-SCR(G,H)

5. Open Problems

• Compute the REC-SCR for Kn displaying k symmetries:

– If H is a dihedral group of size 2n (i.e. maximum size geometric automorphism group), then there are n(n-1)(n-2)(n-3)/24 edge crossings.

– What if we display fewer symmetries?