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The Book Embedding Problem from a SAT-Solving Perspective [GD 2015] Michalis Bekos, Michael Kaufmann, Christian Zielke Universit at T ubingen, Germany The Book Embedding problem Vertices V ordered, on a spine Edges E assigned

  1. SAT formulation General idea: build formula F ( G , p ) via � 1. 1. ensure a proper order of the vertices on the spine σ ( v i , v j ) v i v j � 2. 2. assure that every edge is assigned to one of p pages e i e j χ ( e i , e j ) 3. 3. forbid crossings on each page e i e j forbid χ ( e i , e j ) together with

  2. SAT formulation General idea: build formula F ( G , p ) via � 1. 1. ensure a proper order of the vertices on the spine σ ( v i , v j ) v i v j � 2. 2. assure that every edge is assigned to one of p pages e i e j χ ( e i , e j ) 3. 3. forbid crossings on each page e i e j forbid χ ( e i , e j ) together with σ ( v i , v k ), σ ( v k , v j ), σ ( v j , v l ) v i v k v j v l

  3. SAT formulation General idea: build formula F ( G , p ) via � 1. 1. ensure a proper order of the vertices on the spine σ ( v i , v j ) v i v j � 2. 2. assure that every edge is assigned to one of p pages e i e j χ ( e i , e j ) � 3. 3. forbid crossings on each page e i e j forbid χ ( e i , e j ) together with σ ( v i , v k ), σ ( v k , v j ), σ ( v j , v l ) v i v k v j v l (for every pair of edges 8 forbidden configurations)

  4. SAT formulation General idea: build formula F ( G , p ) via � 1. 1. ensure a proper order of the vertices on the spine σ ( v i , v j ) v i v j � 2. 2. assure that every edge is assigned to one of p pages e i e j χ ( e i , e j ) � 3. 3. forbid crossings on each page solve optimization problem: F ( G , k − 1) is UNSAT , F ( G , k ) is SAT

  5. Experiments Setup: all Rome and North graphs (taken from www.graphdrawing.org)

  6. Experiments Setup: all Rome and North graphs (taken from www.graphdrawing.org) Timeout: 1200 sec

  7. Experiments Setup: all Rome and North graphs (taken from www.graphdrawing.org) Timeout: 1200 sec Rome graphs planar: 3281 nonplanar: 8253 (graphs are very sparse: 0.069)

  8. Experiments Setup: all Rome and North graphs (taken from www.graphdrawing.org) Timeout: 1200 sec Rome graphs planar: 3281 all fit in 2 pages nonplanar: 8253 all fit in 3 pages (graphs are very sparse: 0.069)

  9. Experiments Setup: all Rome and North graphs (taken from www.graphdrawing.org) Timeout: 1200 sec Rome graphs planar: 3281 all fit in 2 pages nonplanar: 8253 all fit in 3 pages (graphs are very sparse: 0.069) North graphs planar: 854 nonplanar: 423 (graphs are denser: 0.13)

  10. Experiments Setup: all Rome and North graphs (taken from www.graphdrawing.org) Timeout: 1200 sec Rome graphs planar: 3281 all fit in 2 pages nonplanar: 8253 all fit in 3 pages (graphs are very sparse: 0.069) North graphs planar: 854 all fit in 2 pages nonplanar: 423 only 344 were solved completely (graphs are denser: 0.13)

  11. Experiments Setup: all Rome and North graphs (taken from www.graphdrawing.org) Timeout: 1200 sec Rome graphs planar: 3281 all fit in 2 pages nonplanar: 8253 all fit in 3 pages (graphs are very sparse: 0.069) North graphs planar: 854 all fit in 2 pages nonplanar: 423 only 344 were solved completely (some graphs have high book thickness) (graphs are denser: 0.13)

  12. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4.

  13. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices

  14. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices Construction: 1. triangulated graph as base (”skeleton”) 1. (not necessarily non-Hamiltonian)

  15. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices Construction: 1. 1. triangulated graph as base (”skeleton”) (not necessarily non-Hamiltonian) 2. 2. augmenting the faces of the base graph by a combination of Stellation Octahedron Creation Graph Insertion

  16. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices Construction: 1. 1. triangulated graph as base (”skeleton”) (not necessarily non-Hamiltonian) 2. 2. augmenting the faces of the base graph by a combination of Stellation Octahedron Creation Graph Insertion

  17. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices Construction: 1. 1. triangulated graph as base (”skeleton”) (not necessarily non-Hamiltonian) 2. 2. augmenting the faces of the base graph by a combination of Stellation Octahedron Creation Graph Insertion

  18. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices Construction: 1. 1. triangulated graph as base (”skeleton”) (not necessarily non-Hamiltonian) 2. 2. augmenting the faces of the base graph by a combination of Stellation Octahedron Creation Graph Insertion

  19. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices Construction: 1. 1. triangulated graph as base (”skeleton”) (not necessarily non-Hamiltonian) 2. 2. augmenting the faces of the base graph by a combination of Stellation Octahedron Creation Graph Insertion

  20. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices Construction: 1. 1. triangulated graph as base (”skeleton”) (not necessarily non-Hamiltonian) 2. 2. augmenting the faces of the base graph by a combination of Stellation Octahedron Creation Graph Insertion

  21. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices Construction: 1. 1. triangulated graph as base (”skeleton”) (not necessarily non-Hamiltonian) 2. 2. augmenting the faces of the base graph by a combination of Stellation Octahedron Creation Graph Insertion

  22. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices Construction: 1. 1. triangulated graph as base (”skeleton”) (not necessarily non-Hamiltonian) 2. 2. augmenting the faces of the base graph by a combination of Stellation Octahedron Creation Graph Insertion

  23. Experiments Hypothesis: There is a (maximal) planar graph whose book thickness is 4. Setup: several maximal planar graphs with 500-700 vertices (all 3 page embeddable) Construction: 1. 1. triangulated graph as base (”skeleton”) (not necessarily non-Hamiltonian) 2. 2. augmenting the faces of the base graph by a combination of Stellation Octahedron Creation Graph Insertion

  24. Experiments - Idea: possibly overcome the limit of ≈ 700 vertices

  25. Experiments - Idea: possibly overcome the limit of ≈ 700 vertices Hypotheses: There is a maximal planar graph G a that has at least one face f a whose edges cannot be on the same page in any book embedding on 3 pages.

  26. Experiments - Idea: possibly overcome the limit of ≈ 700 vertices Hypotheses: There is a maximal planar graph G a that has at least one face f a whose edges cannot be on the same page in any book embedding on 3 pages. test F ( G a , 3) ∪ { ( χ ( e i , e j ) ∧ χ ( e i , e k ) } ∀ f a = ( e i , e j , e k ) ∈ G a

  27. Experiments - Idea: possibly overcome the limit of ≈ 700 vertices Hypotheses: There is a maximal planar graph G a that has at least one face f a whose edges cannot be on the same page in any book embedding on 3 pages. test F ( G a , 3) ∪ { ( χ ( e i , e j ) ∧ χ ( e i , e k ) } ∀ f a = ( e i , e j , e k ) ∈ G a There is a maximal planar graph G c that has at least one face f c whose edges are on the same page in any book embedding on 3 pages.

  28. Experiments - Idea: possibly overcome the limit of ≈ 700 vertices Hypotheses: There is a maximal planar graph G a that has at least one face f a whose edges cannot be on the same page in any book embedding on 3 pages. test F ( G a , 3) ∪ { ( χ ( e i , e j ) ∧ χ ( e i , e k ) } ∀ f a = ( e i , e j , e k ) ∈ G a There is a maximal planar graph G c that has at least one face f c whose edges are on the same page in any book embedding on 3 pages. ∀ f c = ( e i , e j , e k ) ∈ G c test F ( G c , 3) ∪ ( ¬ χ ( e i , e j ) ∨ ¬ χ ( e i , e k ))

  29. Experiments - Idea: possibly overcome the limit of ≈ 700 vertices Hypotheses: There is a maximal planar graph G a that has at least one face f a whose edges cannot be on the same page in any book embedding on 3 pages. test F ( G a , 3) ∪ { ( χ ( e i , e j ) ∧ χ ( e i , e k ) } ∀ f a = ( e i , e j , e k ) ∈ G a There is a maximal planar graph G c that has at least one face f c whose edges are on the same page in any book embedding on 3 pages. ∀ f c = ( e i , e j , e k ) ∈ G c test F ( G c , 3) ∪ ( ¬ χ ( e i , e j ) ∨ ¬ χ ( e i , e k )) G c

  30. Experiments - Idea: possibly overcome the limit of ≈ 700 vertices Hypotheses: There is a maximal planar graph G a that has at least one face f a whose edges cannot be on the same page in any book embedding on 3 pages. test F ( G a , 3) ∪ { ( χ ( e i , e j ) ∧ χ ( e i , e k ) } ∀ f a = ( e i , e j , e k ) ∈ G a There is a maximal planar graph G c that has at least one face f c whose edges are on the same page in any book embedding on 3 pages. ∀ f c = ( e i , e j , e k ) ∈ G c test F ( G c , 3) ∪ ( ¬ χ ( e i , e j ) ∨ ¬ χ ( e i , e k )) f a is outer face G a G c

  31. Experiments - Idea: possibly overcome the limit of ≈ 700 vertices Hypotheses: There is a maximal planar graph G a that has at least one face f a whose edges cannot be on the same page in any book embedding on 3 pages. test F ( G a , 3) ∪ { ( χ ( e i , e j ) ∧ χ ( e i , e k ) } ∀ f a = ( e i , e j , e k ) ∈ G a There is a maximal planar graph G c that has at least one face f c whose edges are on the same page in any book embedding on 3 pages. ∀ f c = ( e i , e j , e k ) ∈ G c test F ( G c , 3) ∪ ( ¬ χ ( e i , e j ) ∨ ¬ χ ( e i , e k )) G a G a G c

  32. Experiments - Idea: possibly overcome the limit of ≈ 700 vertices Hypotheses: There is a maximal planar graph G a that has at least one face f a whose edges cannot be on the same page in any book embedding on 3 pages. test F ( G a , 3) ∪ { ( χ ( e i , e j ) ∧ χ ( e i , e k ) } ∀ f a = ( e i , e j , e k ) ∈ G a There is a maximal planar graph G c that has at least one face f c whose edges are on the same page in any book embedding on 3 pages. ∀ f c = ( e i , e j , e k ) ∈ G c test F ( G c , 3) ∪ ( ¬ χ ( e i , e j ) ∨ ¬ χ ( e i , e k )) G a tested ≈ 284,000 graphs G a G c with 60 to 125 vertices → 0 confirmed Hypotheses

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

  34. Experiments Hypothesis: There is a 1-planar graph whose book thickness is (at least) 4. 5 8 1 4 2 3 6 7

  35. Experiments Hypothesis: There is a 1-planar graph whose book thickness is (at least) 4. 5 8 1 4 1 5 4 2 7 3 8 6 2 3 6 7

  36. Experiments Hypothesis: There is a 1-planar graph whose book thickness is (at least) 5.

  37. Experiments Hypothesis: There is a 1-planar graph whose book thickness is (at least) 5. Setup: all 2,098,675 planar quadrangulations with 25 vertices triconnected; min degree 3 → augment every face with two crossing edges

  38. Experiments Hypothesis: There is a 1-planar graph whose book thickness is (at least) 5. Setup: all 2,098,675 planar quadrangulations with 25 vertices (all 4 page embeddable) triconnected; min degree 3 → augment every face with two crossing edges

  39. Experiments Hypothesis: There is a 1-planar graph whose book thickness is (at least) 5. Setup: all 2,098,675 planar quadrangulations with 25 vertices (all 4 page embeddable) triconnected; min degree 3 → augment every face with two crossing edges

  40. Experiments Hypothesis: There is a 1-planar graph whose book thickness is (at least) 5. Setup: all 2,098,675 planar quadrangulations with 25 vertices (all 4 page embeddable) triconnected; min degree 3 → augment every face with two crossing edges 8312 random optimal 1-planar graphs with 50 − 155 vertices (all 4 page embeddable)

  41. Experiments Hypothesis: There is a 1-planar graph whose book thickness is (at least) 5. Setup: all 2,098,675 planar quadrangulations with 25 vertices (all 4 page embeddable) triconnected; min degree 3 → augment every face with two crossing edges 8312 random optimal 1-planar graphs with 50 − 155 vertices (all 4 page embeddable)

  42. Experiments Hypothesis: There is a (maximal) planar graph, which cannot be embedded in a book of 3 pages, if the subgraphs at each page must be acyclic.

  43. Experiments Hypothesis: There is a (maximal) planar graph, which cannot be embedded in a book of 3 pages, if the subgraphs at each page must be acyclic. (Heath used that for proving book thickness of 3 for planar 3-trees)

  44. Experiments Hypothesis: There is a (maximal) planar graph, which cannot be embedded in a book of 3 pages, if the subgraphs at each page must be acyclic. (Heath used that for proving book thickness of 3 for planar 3-trees) Setup: tested over 15,000 maximal planar graphs with 25 to 80 vertices

  45. Experiments Hypothesis: There is a (maximal) planar graph, which cannot be embedded in a book of 3 pages, if the subgraphs at each page must be acyclic. (Heath used that for proving book thickness of 3 for planar 3-trees) Setup: tested over 15,000 maximal planar graphs with 25 to 80 vertices Timeout: 1200 sec 70.78 % of graphs were solved ≤ 3 minutes 76.37 % of graphs were solved ≤ 20 minutes

  46. Experiments Hypothesis: There is a (maximal) planar graph, which cannot be embedded in a book of 3 pages, if the subgraphs at each page must be acyclic. (Heath used that for proving book thickness of 3 for planar 3-trees) Setup: tested over 15,000 maximal planar graphs with 25 to 80 vertices Timeout: 1200 sec 70.78 % of graphs were solved ≤ 3 minutes 76.37 % of graphs were solved ≤ 20 minutes → additional constraints to force acyclic subgraphs increase runtime

  47. Experiments Hypothesis: (weaker) There is a (maximal) planar graph, which cannot be embedded in a book of 3 pages, if the subgraphs at each page must be acyclic. subgraphs are trees on n − 1 vertices vertices not spanned by trees build a face (w.l.o.g. outer face)

  48. Experiments Hypothesis: (weaker) There is a (maximal) planar graph, which cannot be embedded in a book of 3 pages, if the subgraphs at each page must be acyclic. 16 subgraphs are trees on n − 1 vertices vertices not spanned by trees 15 build a face (w.l.o.g. outer face) 14 12 11 13 9 10 7 8 6 5 3 4 1 2

  49. Experiments Hypothesis: (weaker) There is a (maximal) planar graph, which cannot be embedded in a book of 3 pages, if the subgraphs at each page must be acyclic. 16 subgraphs are trees on n − 1 vertices vertices not spanned by trees 15 build a face (w.l.o.g. outer face) 14 12 11 13 9 → Schnyder decomposition 10 not always possible 7 8 6 5 3 4 1 2

  50. Open problems All Hypothesis are unproven!

  51. Open problems All Hypothesis are unproven! - 1: (maximal) planar graph that requires 4 pages

  52. Open problems All Hypothesis are unproven! - 1: (maximal) planar graph that requires 4 pages - 1a: (maximal) planar graph that requires unicolored face

  53. Open problems All Hypothesis are unproven! - 1: (maximal) planar graph that requires 4 pages - 1a: (maximal) planar graph that requires unicolored face - 1b: (maximal) planar graph that cannot have a unicolored face

  54. Open problems All Hypothesis are unproven! - 1: (maximal) planar graph that requires 4 pages - 1a: (maximal) planar graph that requires unicolored face - 1b: (maximal) planar graph that cannot have a unicolored face - 2: 1-planar graph that requires 5 pages

  55. Open problems All Hypothesis are unproven! - 1: (maximal) planar graph that requires 4 pages - 1a: (maximal) planar graph that requires unicolored face - 1b: (maximal) planar graph that cannot have a unicolored face - 2: 1-planar graph that requires 5 pages - 3: (maximal) planar graph, that requires cyclic subgraphs on pages

  56. Open problems All Hypothesis are unproven! - 1: (maximal) planar graph that requires 4 pages - 1a: (maximal) planar graph that requires unicolored face - 1b: (maximal) planar graph that cannot have a unicolored face - 2: 1-planar graph that requires 5 pages - 3: (maximal) planar graph, that requires cyclic subgraphs on pages Improve the Encoding!

  57. Open problems All Hypothesis are unproven! - 1: (maximal) planar graph that requires 4 pages - 1a: (maximal) planar graph that requires unicolored face - 1b: (maximal) planar graph that cannot have a unicolored face - 2: 1-planar graph that requires 5 pages - 3: (maximal) planar graph, that requires cyclic subgraphs on pages Improve the Encoding! - cope with graphs, that have a high number of vertices and edges

  58. Open problems All Hypothesis are unproven! - 1: (maximal) planar graph that requires 4 pages - 1a: (maximal) planar graph that requires unicolored face - 1b: (maximal) planar graph that cannot have a unicolored face - 2: 1-planar graph that requires 5 pages - 3: (maximal) planar graph, that requires cyclic subgraphs on pages Improve the Encoding! - cope with graphs, that have a high number of vertices and edges - cope with graphs, that have high book thickness

  59. Open problems All Hypothesis are unproven! - 1: (maximal) planar graph that requires 4 pages - 1a: (maximal) planar graph that requires unicolored face - 1b: (maximal) planar graph that cannot have a unicolored face - 2: 1-planar graph that requires 5 pages - 3: (maximal) planar graph, that requires cyclic subgraphs on pages Improve the Encoding! - cope with graphs, that have a high number of vertices and edges - cope with graphs, that have high book thickness Thanks!!!

  60. Random Graph Creation

  61. Random Graph Creation

  62. Random Graph Creation left-to-right sorting

  63. Random Graph Creation left-to-right sorting

  64. Random Graph Creation left-to-right sorting

  65. Random Graph Creation left-to-right sorting

  66. Random Graph Creation left-to-right sorting

  67. Random Graph Creation

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