Generation of oriented matroids using satisfiability solvers Lars - PowerPoint PPT Presentation

Generation of oriented matroids using satisfiability solvers Lars Schewe TU Darmstadt ICMS 2006

Application Triangulations of surfaces [ . . . ] It is well known [ . . . ] that every closed, orientable, topological 2 -manifold M without boundary is imbeddable in R 3 . In analogy to Steinitz’s theorem, one may ask whether every abstract 2 -complex C homeomorphic to such a 2 -manifold M is realizable by a 2 -complex in R 3 , or in any Euclidean space. [ . . . ] If C is simplicial, it is trivially realizable by a 2 -complex in R 5 ; however, no example is known to contradict the conjecture that each simplicial C is realizable by a 2 -complex in R 3 . (Grünbaum, 1967; Exercise 13.2.3)

A counterexample Genus 6, 12 vertices Bokowski and Guedes de Oliveira, 2000 (Model: J. Bokowski, Photograph: N. Hähn)

Results 1. No triangulation of a surface of genus 6 using only 12 vertices admits a polyhedral embedding in R 3 . 2. There exist at least three triangulations of a surface of genus 5 using only 12 vertices that do not admit a polyhedral embedding. 3. For every g ≥ 5 we can construct an infinite family of triangulations of a surface of genus g such that none of these admit a polyhedral embedding in R 3 .

How to prove non-embeddability Oriented matroid approach 3 4 7 7 1 1 6 5 2 2 7 3 4 7

How to prove non-embeddability Oriented matroid approach 3 4 7 7 1 1 Oriented Matroids 6 −→ −→ 5 2 2 7 3 4 7

How to prove non-embeddability Other approaches ◮ T opological obstructions (refined for the PL-case, e.g. van-Kampen-Flores theorem) ◮ Geometric obstructions (further refinement of the criteria above, see Novik) ◮ Geometric arguments (e.g. linking number applied by Brehm) A hybrid strategy is described by Timmreck (to appear). However . . . None of the methods above was sucessfully applied to decide a genus 6, 12 vertex example.

Triangulations 3 4 7 7 1 1 6 5 2 2 7 7 3 4

Triangulations 3 4 7 7 ◮ every edge is contained in 1 1 6 two triangles 5 2 2 7 7 3 4

Triangulations 3 4 7 7 ◮ every edge is contained in 1 1 6 two triangles ◮ every link of a vertex is a 5 2 2 cycle 7 7 3 4

Bounds for combinatorial triangulations Theorem (Jungerman and Ringel, 1980) Let S be a surface of genus g (g � = 2 ). Then there exists a triangulation of S with n vertices, if and only if: � n − 3 � ≥ 6 g 2 That means that with n vertices one can triangulate a surface of genus O ( n 2 ) .

Number of combinatorial triangulations # g n min 0 4 1 1 7 1 2 10 865 (Lutz, 2003) 3 10 20 (Lutz, 2003) 4 11 821 (Lutz, 2005) 5 12 751 593 (Sulanke, 2005) 6 12 59 (Altshuler et al., 1996)

Polyhedral Embeddings General construction Theorem (McMullen, Schulz, Wills, 1983) There exist triangulations using n vertices of surfaces of genus O ( n log n ) , which admit a polyhedral embedding.

Polyhedral Embeddings Current knowledge for g ≤ 5 g = 0 all triangulations are realizable (Steinitz) 1 ≤ g ≤ 4 all minimal vertex triangulations are realizable (genus 1, Cszásár, 1949; genus 2, Lutz and Bokowski, 2005; genus 3, 4: Hougardy, Lutz, Zelke, 2005) g = 5 Some minimal vertex triangulations are realizable (Hougardy, Lutz, Zelke, 2005)

Polyhedral Embeddings Methods Methods for small cases ◮ direct construction (Bokowski, Brehm) ◮ random coordinates (Lutz, 2005) ◮ small coordinates (Hougardy, Lutz, Zelke, 2005) ◮ local search (Hougardy, Lutz, Zelke, 2005)

Oriented matroids Definition Let p 1 , · · · , p n ∈ R d and � p i = ( 1 , p i ) ∈ R d + 1 . Then we call χ ( i 1 , · · · , i d + 1 ) : = sgndet ( � p i 1 , · · · , � p i d + 1 ) the chirotope of the point set. A chirotope is called uniform , if χ ( i 1 , · · · , i d + 1 ) � = 0, where the i j are pairwise disjoint.

Oriented matroids 4 Example: d = 2 ◮ p i , p j , p k collinear ⇔ sgndet ( � p i , � p j , � p k ) = 0 ◮ p i , p j , p k ccw 3 1 ⇔ sgndet ( � p i , � p j , � p k ) = + 5 2

Oriented matroids 4 not part of the convex hull 3 1 5 2

Oriented matroids 4 part of the convex hull 3 1 5 2

Oriented matroids For a combinatorial model we use a combinatorial version of the Grassmann-Plücker relations : r + 1 � ( − 1 ) j det ( x 1 , · · · , x j − 1 , x j + 1 , · · · , x r + 1 ) det ( x j , y 1 , · · · , y r − 1 ) = 0 j = 1

Oriented matroids For a combinatorial model we use a combinatorial version of the Grassmann-Plücker relations ( r = 2): det ( x 2 , x 3 ) det ( x 1 , y 1 ) − det ( x 1 , x 3 ) det ( x 2 , y 1 ) + det ( x 1 , x 2 ) det ( x 3 , y 1 ) = 0

Oriented matroids For a combinatorial model we use a combinatorial version of the Grassmann-Plücker relations ( r = 2): χ ( b , c ) χ ( a , d ) − χ ( a , c ) χ ( b , d ) + χ ( a , b ) χ ( c , d )

Oriented matroids For a combinatorial model we use a combinatorial version of the Grassmann-Plücker relations ( r = 2): χ ( b , c ) χ ( a , d ) = 0 − χ ( a , c ) χ ( b , d ) = 0 + χ ( a , b ) χ ( c , d ) = 0

Oriented matroids For a combinatorial model we use a combinatorial version of the Grassmann-Plücker relations ( r = 2): χ ( b , c ) χ ( a , d ) = + − χ ( a , c ) χ ( b , d ) = − + χ ( a , b ) χ ( c , d )

Oriented matroids For a combinatorial model we use a combinatorial version of the Grassmann-Plücker relations ( r = 2): χ ( b , c ) χ ( a , d ) − χ ( a , c ) χ ( b , d ) = + + χ ( a , b ) χ ( c , d ) = −

Oriented matroids For a combinatorial model we use a combinatorial version of the Grassmann-Plücker relations ( r = 2): χ ( b , c ) χ ( a , d ) − χ ( a , c ) χ ( b , d ) + χ ( a , b ) χ ( c , d ) The higher-dimensional case can be reduced to the linear situation via contraction

Definition Let E be a finite set, r ∈ N and χ : E r → { + , 0 , − } . We call M = ( E , χ ) an oriented matroid, if: (B1) The mapping χ is alternating.

Definition Let E be a finite set, r ∈ N and χ : E r → { + , 0 , − } . We call M = ( E , χ ) an oriented matroid, if: (B1) The mapping χ is alternating. (B2) The set B = {{ x 1 , · · · , x r } | χ ( x 1 , · · · , x r ) � = 0 } is the set of bases of a matroid.

Definition Let E be a finite set, r ∈ N and χ : E r → { + , 0 , − } . We call M = ( E , χ ) an oriented matroid, if: (B1) The mapping χ is alternating. (B2) The set B = {{ x 1 , · · · , x r } | χ ( x 1 , · · · , x r ) � = 0 } is the set of bases of a matroid. (B3) For all σ ∈ � E � and all subsets { x 1 , . . . , x 4 } ⊆ E \ σ r − 2 one of the two following conditions holds: ◮ s 1 = s 2 = s 3 = 0 ◮ { s 1 , s 2 , s 3 } ⊇ { − , + } Here we set: s 1 = χ ( σ, x 1 , x 2 ) χ ( σ, x 3 , x 4 ) s 2 = − χ ( σ, x 1 , x 3 ) χ ( σ, x 2 , x 4 ) s 3 = χ ( σ, x 1 , x 4 ) χ ( σ, x 2 , x 3 )

Generating oriented matroids ◮ Hyperline sequences (Bokowski, Guedes de Oliveira, 2000) ◮ Cocircuit graphs (Finschi and Fukuda, 2002) ◮ Chirotope (Bremner, 2004)

Satisfiability: CNF x 1 x 2 x 3 ◮ Variables

Satisfiability: CNF x 1 x 2 x 3 ¬ x 1 ¬ x 2 ¬ x 3 ◮ Literals

Satisfiability: CNF x 1 ∨ x 2 ∨ x 3 ¬ x 1 ∨ ¬ x 2 ∨ ¬ x 3 ¬ x 1 ∨ x 2 x 2 ∨ ¬ x 3 x 1 ∨ ¬ x 2 ∨ ¬ x 3 ◮ Clauses x 1 ∨ ¬ x 2 ∨ x 3

Satisfiability: CNF � x 1 � ∨ x 2 ∨ x 3 � ¬ � ◮ CNF ∧ x 1 ∨ ¬ x 2 ∨ ¬ x 3 � ¬ � ∧ x 1 ∨ x 2 � x 2 � ∧ ∨ ¬ x 3 � x 1 � ∧ ∨ ¬ x 2 ∨ ¬ x 3 � x 1 � ∧ ∨ ¬ x 2 ∨ x 3

First steps towards the model Theorem (Peirce) Given: f : { 0 , 1 } n → { 0 , 1 } . Then:   � � �   f ( x ) =  x i ∨ ¬ x i  v ∈ { z | f ( z )= 0 } i ∈ { j | v j = 0 } i ∈ { j | v j = 1 }

First steps towards the model χ ( a ) χ ( b ) = + ⇒ χ ( c ) χ ( d ) = −

First steps towards the model χ ( a ) χ ( b ) = + ⇒ χ ( c ) χ ( d ) = − ¬ x a ¬ x b ¬ x c ¬ x d

First steps towards the model χ ( a ) χ ( b ) = + ⇒ χ ( c ) χ ( d ) = − ¬ x a ¬ x b ¬ x c ¬ x d ¬ x a ¬ x b x c x d

Size of the model In general (and before preprocessing) ◮ 1 variable per basis ◮ 16 clauses per Grassmann-Plücker relation ◮ 2 clauses per intersection condition Genus 6 (after preprocessing) ◮ 494 variables ◮ between 225021 and 225148 clauses

SAT solvers Solvers ◮ ZChaff (Fu, Mahajan, Malik, 2004) ◮ MiniSat (Eén and Sörensson, 2003–) Preprocessor ◮ SatELite (Eèn and Biere, 2005)

Running times Oriented matroids do not exist Brehm’s triangulation of the Möbius’ strip ≈ 2s Genus 6, 12 vertices between 1h and 2h Genus 5, 12 vertices between 1h and 2h Oriented matroids exist Genus 1, 7 vertices ≈ 18s (2772 oriented matroids) Genus 5, 12 vertices ≈ 86s (until the first solution); ≈ 40min (for all)

Results 1. No triangulation of a surface of genus 6 using only 12 vertices admits a polyhedral embedding in R 3 . 2. There exist at least three triangulations of a surface of genus 5 using only 12 vertices that do not admit a polyhedral embedding. 3. For every g ≥ 5 we can construct an infinite family of triangulations of a surface of genus g such that none of these admit a polyhedral embedding in R 3 .