Using SAT Solvers in Combinatorics and Geometry Manfred Scheucher 1
Boolean satisfiability problem • Given Boolean formula, is there an assignment such that the formula is true? 2
Boolean satisfiability problem • Given Boolean formula, is there an assignment such that the formula is true? • NP-complete, but quite good heuristics 2
Why to use SAT Solvers? • Prove/disprove existence of structure usually faster 3
Why to use SAT Solvers? • Prove/disprove existence of structure • Find (counter)examples • Counting occurences • Induction base • Induction step 3
Today’s Topics • L-shaped Point Set Embeddings of Trees (with Torsten M¨ utze) • Orthogonal Symmetric Chain Decompositions (with Karl D¨ aubel, Sven J¨ ager, and Torsten M¨ utze) • (Disjoint) Holes in Point Sets • Universal Point Sets for Planar Graphs (with Hendrik Schrezenmaier and Raphael Steiner) 4
L-Shaped Point Set Embeddings joint work with Torsten M¨ utze arXiv:1807.11043 5
L-Shaped Point Set Embeddings T . . . tree on n vertices P . . . set of m points • vertices of T drawn as points of P • edges drawn as unions of two axis parallel line segments 6
L-Shaped Point Set Embeddings f ( T ) . . . minimum number m s.t. tree T admits a planar L -shaped embedding in any set of m points f d ( n ) := max f ( T ) T : tree on n vertices max. deg. ∆( T ) ≤ d 7
L-Shaped Point Set Embeddings f ( T ) . . . minimum number m s.t. tree T admits a planar L -shaped embedding in any set of m points f d ( n ) := max f ( T ) T : tree on n vertices max. deg. ∆( T ) ≤ d • f 4 ( n ) ≤ n 2 [Di Giacomo, Frati, Fulek, Grilli, Krug ’13] • f 4 ( n ) ≤ O ( n 1 . 58 ) [S.’15, Aichholzer-Hackl-S.’16] • f 3 ( n ) ≤ O ( n 1 . 22 ) , f 4 ( n ) ≤ O ( n 1 . 55 ) [Biedl et al.’17] • no non-trivial lower bound ( ≥ n is trivial) 7
SAT Model • T . . . tree on vertices { v 1 , . . . , v n } • P . . . point set { P 1 , . . . , P n } • formulate Boolean satisfiability instance: ∃ solution iff. T admits an L-shaped embedding in P 8
SAT Model: Variables • M i,j . . . vertex v i is mapped to point P j • H a,b . . . edge ab is connected horizontally to a 9
SAT Model: Clauses • Injective mapping V to P every vertex v i has to be mapped: � M i,j j no two vertices v i 1 , v i 2 are mapped to the same point: ¬ M i 1 ,j ∨ ¬ M i 2 ,j 10
SAT Model: Clauses • Injective mapping V to P • L-shaped edges: ab connects either vertically or horizontally to a (and b ) H a,b ∨ H b,a , ¬ H a,b ∨ ¬ H b,a a b 10
SAT Model: Clauses • Injective mapping V to P • L-shaped edges: ab connects either vertically or horizontally to a (and b ) • No overlapping edges 10
SAT Model: Clauses • Injective mapping V to P • L-shaped edges: ab connects either vertically or horizontally to a (and b ) • No overlapping edges order of points is important p r q 10
SAT Model: Clauses • Injective mapping V to P • L-shaped edges: ab connects either vertically or horizontally to a (and b ) • No overlapping edges only depends on point set If p left of q and r , only depends on variables and v a → p , v b → q , v c → r , then not both edges ab and ac horizonally connected p r q 10
SAT Model: Clauses • Injective mapping V to P • L-shaped edges: ab connects either vertically or horizontally to a (and b ) • No overlapping edges only depends on point set If p left of q and r , only depends on variables and v a → p , v b → q , v c → r , then not both edges ab and ac horizonally connected For each three points p, q, r with p left of q and r : ¬ M a,p ∨ ¬ M b,q ∨ ¬ M c,r ∨ ¬ H a,b ∨ ¬ H a,c 10
SAT Model: Clauses • Injective mapping V to P • L-shaped edges: ab connects either vertically or horizontally to a (and b ) • No overlapping edges • No crossing edges 10
Results Theorem: Every tree on n ≤ 12 vertices admits an L-shaped embedding in every set of n points. Theorem: T 13 has no L-shaped embedding in P 13 . T 13 P 13 11
Results Theorem: Every tree on n ≤ 12 vertices admits an L-shaped embedding in every set of n points. Theorem: T 13 has no L-shaped embedding in P 13 . • Further examples for n ∈ { 13 , 14 , 16 , 17 , 18 , 19 , 20 } • If cyclic order fixed, infinite family . . . 11
And now for something completely different 12
Orthogonal Chain Decompositions joint work with Karl D¨ aubel, Sven J¨ ager, and Torsten M¨ utze arXiv:1810.09847 13
Orthogonal Chain Decompositions • Boolean lattice Q n n � � can be partitioned into chains [Dilworth ’50] ⌊ n/ 2 ⌋ 14
Orthogonal Chain Decompositions { 1 , 2 , 3 , 4 } { 1 , 2 , 3 } { 1 , 2 , 4 } { 1 , 3 , 4 } { 2 , 3 , 4 } { 1 , 2 } { 1 , 3 } { 1 , 4 } { 2 , 3 } { 2 , 4 } { 3 , 4 } { 1 } { 2 } { 3 } { 4 } ∅ 14
Orthogonal Chain Decompositions • Boolean lattice Q n n � � can be partitioned into chains [Dilworth ’50] ⌊ n/ 2 ⌋ • orthogonal chain-decompositions: any two chains from two decompositions have at most one element in common 14
Orthogonal Chain Decompositions • ∃ two orthogonal CDs in Q n ≥ 2 [Shearer-Kleitman ’79] • moreover, they conjectured ⌊ n/ 2 ⌋ + 1 15
Orthogonal Chain Decompositions • ∃ two orthogonal CDs in Q n ≥ 2 [Shearer-Kleitman ’79] • moreover, they conjectured ⌊ n/ 2 ⌋ + 1 • ∃ three orthogonal CDs in Q n ≥ 24 [Spink ’17] 15
Orthogonal Chain Decompositions • ∃ two orthogonal CDs in Q n ≥ 2 [Shearer-Kleitman ’79] • moreover, they conjectured ⌊ n/ 2 ⌋ + 1 • ∃ three orthogonal CDs in Q n ≥ 24 [Spink ’17] Proof Idea: 1. (specific) orthogonal CDs for Q 5 and Q 7 2. Lemma ” Q a , Q b ⇒ Q a + b ” 15
Orthogonal Chain Decompositions • ∃ two orthogonal CDs in Q n ≥ 2 [Shearer-Kleitman ’79] • moreover, they conjectured ⌊ n/ 2 ⌋ + 1 • ∃ three orthogonal CDs in Q n ≥ 24 [Spink ’17] n 1 2 3 4 5 6 7 8 9 10 11 almost-orth. 1 2 2 2 3 3* 4* 3* 3* 3 4* ⌊ n/ 2 ⌋ + 1 1 2 2 3 3 4 4 5 5 6 6 found via SAT solvers! 15
Orthogonal Chain Decompositions • ∃ two orthogonal CDs in Q n ≥ 2 [Shearer-Kleitman ’79] • moreover, they conjectured ⌊ n/ 2 ⌋ + 1 • ∃ three orthogonal CDs in Q n ≥ 24 [Spink ’17] n 1 2 3 4 5 6 7 8 9 10 11 almost-orth. 1 2 2 2 3 3* 4* 3* 3* 3 4* ⌊ n/ 2 ⌋ + 1 1 2 2 3 3 4 4 5 5 6 6 Theorem: ∃ four orthogonal CDs in Q n ≥ 60 . [D¨ aubel-J¨ ager-M¨ utze-S. ’18] 15
SAT Instance • Problem: naive CNF formulation much too big 16
SAT Instance • Problem: naive CNF formulation much too big • use symmetries 16
SAT Instance { } 1 , 2 , 3 , 4 , 5 { } { } { } { } { } 1 , 2 , 3 , 4 1 , 2 , 3 , 5 1 , 2 , 4 , 5 1 , 3 , 4 , 5 2 , 3 , 4 , 5 { } { } { } { } { } { } { } { } { } { } 1 , 2 , 3 1 , 2 , 4 1 , 2 , 5 1 , 3 , 4 1 , 3 , 5 1 , 4 , 5 2 , 3 , 4 2 , 3 , 5 2 , 4 , 5 3 , 4 , 5 { } { } { } { } { } { } { } { } { } { } 1 , 2 1 , 3 1 , 4 1 , 5 2 , 3 2 , 4 2 , 5 3 , 4 3 , 5 4 , 5 { } { } { } { } { } 1 2 3 4 5 ∅ 16
SAT Instance �{ 1 , 2 , 3 , 4 , 5 }� { 1 , 2 , 3 , 4 , 5 } { 1 , 2 , 3 , 4 { 1 , 2 , 3 , 5 } { 1 , 2 , 4 , 5 } { 1 , 3 , 4 , 5 } { 2 , 3 , 4 , 5 } } �{ 1 , 2 , 3 , 4 }� �{ 1 , 3 , 4 }� { 1 , 2 , 3 { 1 , 2 , 4 } { 1 , 2 , 5 } { 1 , 3 , 4 } { 1 , 3 , 5 } { 1 , 4 , 5 } { 2 , 3 , 4 } { 2 , 3 , 5 } { 2 , 4 , 5 } { 3 , 4 , 5 } } �{ 1 , 2 , 3 }� �{ 1 , 2 }� �{ 1 , 3 }� { } { } { } { } { } { } { } { } { } { } 1 , 2 1 , 3 1 , 4 1 , 5 2 , 3 2 , 4 2 , 5 3 , 4 3 , 5 4 , 5 �{ 1 }� { } { } { } { } { } 1 2 3 4 5 �∅� ∅ 16
SAT Instance • Problem: naive CNF formulation much too big • use symmetries • iteratively add clauses to “correct errors” (incremental solving supported by minisat/glucose) 16
And now for something more completely different 17
Classical Erd˝ os–Szekeres • Given n points in the plane in general position, is there subset size k in convex position (” k -gon”)? 18
Classical Erd˝ os–Szekeres • Given n points in the plane in general position, is there subset size k in convex position (” k -gon”)? 7-gon 18
Classical Erd˝ os–Szekeres • Given n points in the plane in general position, is there subset size k in convex position (” k -gon”)? Theorem. 2 k − 2 + 1 ≤ g ( k ) ≤ � 2 k − 4 � . [Erd˝ os–Szekeres ’35] k − 2 equality conjectured by Szekeres, Erd˝ os offered 500$ for a proof 18
Classical Erd˝ os–Szekeres • Given n points in the plane in general position, is there subset size k in convex position (” k -gon”)? Theorem. 2 k − 2 + 1 ≤ g ( k ) ≤ � 2 k − 4 � . [Erd˝ os–Szekeres ’35] k − 2 . . . Theorem. g ( k ) ≤ 2 k + o ( k ) . [Suk ’17] 18
Classical Erd˝ os–Szekeres • Given n points in the plane in general position, is there subset size k in convex position (” k -gon”)? Theorem. 2 k − 2 + 1 ≤ g ( k ) ≤ � 2 k − 4 � . [Erd˝ os–Szekeres ’35] k − 2 . . . Theorem. g ( k ) ≤ 2 k + o ( k ) . [Suk ’17] Known: g (4) = 5 , g (5) = 9 , g (6) = 17 computer assisted proof, 1500 CPU hours [Szekeres–Peters ’06] 18
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