from a sat-solving perspective the book embedding problem

Michalis Bekos, Michael Kaufmann, Christian Zielke

Universitat Tubingen, Germany

The Book Embedding Problemfrom a SAT-Solving Perspective

[GD 2015]

The Book Embedding problem

Vertices V ordered, on a spine→

Edges E assigned to p pages→

Pages half planes, delimited by spineplanar

→





→

goal: minimize p - book thickness





→

1

2

3 4 5 6 7

8 9

10 11






→

1

2

3 4 5 6 7

8 9

10 11

1 7 6 11 5 10 2 3 4 8 9



Known results:

[Bernhard & Kainen, 1979]




→

1

2

3 4 5 6 7

8 9

10 11

1 7 6 11 5 10 2 3 4 8 9


bt(G ) = 1⇔ G is outerplanar, bt(G ) = 2⇔ G is subhamiltonian


all planar graphs can be embedded on 4 pages

Known results:

[Yannakakis, 1989]





→

(no graph needing 4 pages is known)

1

2

3 4 5 6 7

8 9

10 11

1 7 6 11 5 10 2 3 4 8 9




all planar graphs can be embedded on 4 pages

Known results:

[Yannakakis, 1989]


1-planar graphs have constant book thickness

[Bekos et al., 2015]




→

(no graph needing 4 pages is known)

(39)

1

2

3 4 5 6 7

8 9

10 11

1 7 6 11 5 10 2 3 4 8 9



SAT solving

Boolean variables ”sun is shining”, ”road is wet”, α, β, ...

SAT solving

Boolean variables

clauses

”sun is shining”, ”road is wet”, α, β, ...

conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...

SAT solving

Boolean variables

clauses


conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...formula

SAT solving

logical operations

→ implications: ”if ... then ...”

↔ equivalence: ”if and only if”

Boolean variables

clauses


conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...formula

SAT solving

logical operations



”forbid”particular settings

Boolean variables

clauses



(α ∨ ¬β) forbids ¬α,β

formula

SAT solving

logical operations




Boolean variables

clauses




SAT? assignment to variables that satisfies all clauses

formula

SAT solving

logical operations




Boolean variables

clauses




SAT? assignment to variables that satisfies all clauses

formula

use SAT solver as black box / oracle

SAT solving

logical operations




Boolean variables

clauses




formula

Applications of SAT:

[Zeranski & Chimani, 2012]

Upward planarity via SAT

SAT solving

logical operations




Boolean variables

clauses




formula

Applications of SAT:

[Biedl et al, 2013]

SAT for grid-based graph problems (pathwidth, vis. representation, ...)

[Zeranski & Chimani, 2012]

Upward planarity via SAT

SAT formulation

General idea:

1. ensure a proper order of the vertices on the spine1.

build formula F(G , p) via

SAT formulation

General idea:


vi


SAT formulation

General idea:


vi vj

σ(vi , vj) : vi is left of vj on spine

Variables:


SAT formulation

General idea:


vi vj


Variables:

Rules:


SAT formulation

General idea:


vi vj


Variables:

Rules:

Antisymmetry: σ(vi , vj)↔ ¬σ(vj , vi )


SAT formulation

General idea:


vi vjvj


Variables:

Rules:



SAT formulation

General idea:


vi vj


Variables:

Rules:


Transitivity: σ(vi , vj) ∧ σ(vj , vk)→ σ(vi , vk)


SAT formulation

General idea:


vi vj vk


Variables:

Rules:


Transitivity: σ(vi , vj) ∧ σ(vj , vk)→ σ(vi , vk)


SAT formulation

General idea:


vi vj

X

σ(vi , vj)


SAT formulation

General idea:

1.

2.

ensure a proper order of the vertices on the spine

assure that every edge is assigned to one of p pages

1.

2.

vi vj

X

σ(vi , vj)

Variables:φp(ei ) : edge ei is assigned to page p


SAT formulation

General idea:

1.

2.



1.

2.

vi vj

X

σ(vi , vj)


Rules:


SAT formulation

General idea:

1.

2.



1.

2.

vi vj

X

σ(vi , vj)


Rules:≥ 1 page: φ1(ei ) ∨ φ2(ei ) ∨ . . . ∨ φp(ei )


SAT formulation

General idea:

1.

2.



1.

2.

vi vj

X

σ(vi , vj)



χ(ei , ej) : edges ei and ej are assigned to same page


SAT formulation

General idea:

1.

2.



1.

2.

vi vj

X

σ(vi , vj)



χ(ei , ej) : edges ei and ej are assigned to same page

same page: (φk(ei ) ∧ φk(ej))→ χ(ei , ej)


SAT formulation

General idea:

1.

2.



1.

2.

vi vj

X

X

σ(vi , vj)

χ(ei , ej)ei ej


SAT formulation

General idea:

1.

2.

3.



1.

2.

forbid crossings on each page3.

vi vj

X

X

σ(vi , vj)

χ(ei , ej)ei ej


SAT formulation

General idea:

1.

2.

3.



1.

2.


vi vj

X

X

σ(vi , vj)

χ(ei , ej)ei ej

forbid χ(ei , ej) together withei ej


SAT formulation

General idea:

1.

2.

3.



1.

2.


vi vj

X

X

σ(vi , vj)

χ(ei , ej)ei ej

vi vjvk vl


σ(vi , vk),σ(vk , vj),σ(vj , vl)


SAT formulation

General idea:

1.

2.

3.



1.

2.


vi vj

X

X

X

σ(vi , vj)

χ(ei , ej)ei ej

vi vjvk vl


σ(vi , vk),σ(vk , vj),σ(vj , vl)(for every pair of edges 8 forbidden configurations)


SAT formulation

General idea:

1.

2.

3.



1.

2.


vi vj

X

X

X

σ(vi , vj)

χ(ei , ej)ei ej


solve optimization problem: F(G , k − 1) is UNSAT, F(G , k) is SAT

Experiments

Setup:

all Rome and North graphs (taken from www.graphdrawing.org)

Experiments

Setup:


Timeout: 1200 sec

Experiments

Setup:


Rome graphs

planar: 3281

nonplanar: 8253(graphs are very sparse: 0.069)

Timeout: 1200 sec

Experiments

Setup:


Rome graphs

planar: 3281 all fit in 2 pages

nonplanar: 8253 all fit in 3 pages(graphs are very sparse: 0.069)

Timeout: 1200 sec

Experiments

Setup:


Rome graphs



North graphs

planar: 854

nonplanar: 423

Timeout: 1200 sec

(graphs are denser: 0.13)

Experiments

Setup:


Rome graphs



North graphs


nonplanar: 423 only 344 were solved completely

Timeout: 1200 sec

(graphs are denser: 0.13)

Experiments

Setup:


Rome graphs



North graphs


nonplanar: 423 only 344 were solved completely

Timeout: 1200 sec

(graphs are denser: 0.13)(some graphs have high book thickness)

Experiments

Hypothesis:

There is a (maximal) planar graph whose book thickness is 4.

Experiments

Hypothesis:


Setup:

several maximal planar graphs with 500-700 vertices

Experiments

Hypothesis:


Setup:


Construction:

1. triangulated graph as base (”skeleton”)1. (not necessarily non-Hamiltonian)

Experiments

Hypothesis:


Setup:


Construction:

1.

2.

triangulated graph as base (”skeleton”)

augmenting the faces of the base graph by a combination of

1.

2.

(not necessarily non-Hamiltonian)

Stellation Octahedron Creation Graph Insertion

Experiments

Hypothesis:


Setup:


Construction:

1.

2.

triangulated graph as base (”skeleton”)

augmenting the faces of the base graph by a combination of

1.

2.

(not necessarily non-Hamiltonian)

Stellation Octahedron Creation Graph Insertion

(all 3 page embeddable)

Experiments

- Idea: possibly overcome the limit of ≈ 700 vertices

Experiments


Hypotheses:

There is a maximal planar graph Ga that has at least one face fa whoseedges cannot be on the same page in any book embedding on 3 pages.

Experiments


Hypotheses:


test F(Ga, 3) ∪ {(χ(ei , ej) ∧ χ(ei , ek)}∀fa = (ei , ej , ek) ∈ Ga

Experiments


Hypotheses:


There is a maximal planar graph Gc that has at least one face fc whoseedges are on the same page in any book embedding on 3 pages.


Experiments


Hypotheses:



∀fc = (ei , ej , ek) ∈ Gc

test F(Gc , 3) ∪ (¬χ(ei , ej) ∨ ¬χ(ei , ek))


Experiments


Hypotheses:






Gc

Experiments


Hypotheses:






Gc Gafa is outer face

Experiments


Hypotheses:






GcGa

Ga

Experiments


Hypotheses:






GcGa

Ga tested ≈ 284,000 graphswith 60 to 125 vertices→ 0 confirmed Hypotheses

Experiments

Hypothesis:

There is a 1-planar graph whose book thickness is (at least) 4.

Experiments

Hypothesis:


1

2 3

4

5

6 7

8

Experiments

Hypothesis:


1

2 3

4

5

6 7

8

1 2 345 67 8

Experiments

Hypothesis:


Experiments

Hypothesis:

Setup:

all 2,098,675 planar quadrangulations with 25 vertices


triconnected; min degree 3 → augment every face with two crossing edges

Experiments

Hypothesis:

Setup:

all 2,098,675 planar quadrangulations with 25 vertices (all 4 page embeddable)



Experiments

Hypothesis:

Setup:




8312 random optimal 1-planar graphs with 50− 155 vertices(all 4 page embeddable)

Experiments

Hypothesis:

There is a (maximal) planar graph, which cannot be embeddedin a book of 3 pages, if the subgraphs at each page must be acyclic.

Experiments

Hypothesis:


(Heath used that for proving book thickness of 3 for planar 3-trees)

Experiments

Hypothesis:


Setup:

tested over 15,000 maximal planar graphs with 25 to 80 vertices


Experiments

Hypothesis:


Setup:


70.78 % of graphs were solved ≤ 3 minutes

Timeout: 1200 sec



Experiments

Hypothesis:


Setup:



Timeout: 1200 sec


→ additional constraints to force acyclic subgraphs increase runtime


Experiments

Hypothesis:


(weaker)

subgraphs are trees on n − 1 vertices

vertices not spanned by treesbuild a face(w.l.o.g. outer face)

Experiments

Hypothesis:


(weaker)



1

16

2

3 45

6

7 8

9

15

10

1112 13

14

Experiments

Hypothesis:


(weaker)



1

16

2

3 45

6

7 8

9

15

10

1112 13

14

→ Schnyder decompositionnot always possible

Open problems

All Hypothesis are unproven!

Open problems


- 1: (maximal) planar graph that requires 4 pages

Open problems



- 1a: (maximal) planar graph that requires unicolored face

Open problems




- 1b: (maximal) planar graph that cannot have a unicolored face

Open problems





- 2: 1-planar graph that requires 5 pages

Open problems






- 3: (maximal) planar graph, that requires cyclic subgraphs on pages

Open problems







Improve the Encoding!

Open problems








- cope with graphs, that have a high number of vertices and edges

Open problems









- cope with graphs, that have high book thickness

Open problems


Thanks!!!








- cope with graphs, that have high book thickness

Random Graph Creation


left-to-right sorting

Experiments

Hypothesis:

Setup:


Construction:

1. start with cube1.




8312 random optimal 1-planar graphs with 50− 155 vertices

Experiments

Hypothesis:

Setup:


Construction:

1.

2.

start with cube

apply one of the two operations defined by

1.

2.





[Suzuki, 2006]

Experiments

Hypothesis:

Setup:




8312 random optimal 1-planar graphs with 50− 155 vertices(all 4 page embeddable)

Runtime Rome

Runtime North

from a sat-solving perspective the book embedding problem

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