Making Path-Consistency Stronger for SAT Pavel Surynek Faculty of Mathematics and Physics Charles University in Prague Czech Republic

Outline of the talk Motivation SAT problems Constraint model of SAT and Path-consistency Modification of Path-consistency Complexity and propagation algorithm Experimental evaluation  propagation strength comparison  comparison on SAT preprocessing CSCLP 2008 Pavel Surynek

Motivation Local consistencies  local inference often too weak for SAT  arc-consistency, path-consistency, i,j-consistency insignificant gain in comparison with unit-propagation expensive propagation w.r.t. inference strength Global consistencies (global constraints)  strong global inference often significant simplification of the problem  no explicit global constraints in SAT Consistency based on structural properties  exploit structure for global propagation in SAT CSCLP 2008 Pavel Surynek

Boolean Satisfaction Problem (SAT) A Boolean formula is given - variables can take either a value true or false The task is to find valuation of variables such that the formula is satisfied  or decide that no such valuation exists Conjunctive normal form (CNF) - standard form of the input formula  variables: x 1,x 2,x 3,...  literals: x 1,  x 1,x 2,  x 2,... variable or its negation  clauses: (x 1   x 2   x 3 )... disjunction of literals  formula: (x 1   x 2 )  (x 1  x 2   x 3 )... conjunction of clauses CSCLP 2008 Pavel Surynek example: x = true y = false example: (  x   y)  (x   y) example: p cnf

SAT as CSP: Literal encoding model (X,D,C)  X... variables ↔ clauses  D... variable domains ↔ literals  C... constraints ↔ values standing for complementary literals are forbidden Consider SAT CSP model as an undirected graph  vertices ↔ values in domains, edges ↔ allowed pairs of values (not all shown in the example) V (  x1   x2) x1x1 x2x2 x1x1 x2x2 x2x2 x3x3 x2x2 x3x3 x3x3 x1x1 x3x3 x1x1 x1Vx2x1Vx2 x1Vx2x1Vx2 x2Vx3x2Vx3 x2Vx3x2Vx3 x3Vx1x3Vx1 x3Vx1x3Vx1     Constraint Model for SAT CSCLP 2008 Pavel Surynek example: V (  x1   x2),V (x1  x2),... example: V (  x1   x2)  {  x 1,  x 2 } example: V (  x1   x2) =  x 1 and V (x1  x2) = x 1 is forbidden

Let us have a sequence of variables (path)  pair of values is path-consistent w.r.t. to the sequence if there is a path between them in the graph that uses the edges between neighboring variables in the sequence Ignores constraints not between variables neighboring in the sequence of variables x1x1 x2x2 x1x1 x2x2 x2x2 x3x3 x2x2 x3x3 x3x3 x1x1 x3x3 x1x1 x1Vx2x1Vx2 x1Vx2x1Vx2 x2Vx3x2Vx3 x2Vx3x2Vx3 x3Vx1x3Vx1 x3Vx1x3Vx1     Path-Consistency for SAT: a graph interpretation CSCLP 2008 Pavel Surynek

Modified Path-Consistency for SAT Deduce more information from constraints  decompose values into disjoint sets (called layers... L 1, L 2,...)  deduce more information from constraints - calculate maximum numbers of visits by a path in layers Stronger restriction on paths → stronger propagation CSCLP 2008 Pavel Surynek x1x1 x2x2 x1x1 x2x2 x2x2 x3x3 x2x2 x3x3 x3x3 x1x1 x3x3 x1x1 x1Vx2x1Vx2 x1Vx2x1Vx2 x2Vx3x2Vx3 x2Vx3x2Vx3 x3Vx1x3Vx1 x3Vx1x3Vx1     L1L1 L2L2 path ending in this vertex must not visit L 1 more than two times

NP-Completeness of Modified Path-Consistency Enforcing modified path-consistency is difficult (NP-complete) Theorem: Existence of a Hamiltonian path in a graph is reducible to the existence of the path respecting maximum numbers of visits. Main idea of the proof: G=(V,E), where V={v 1,v 2,...,v n } CSCLP 2008 Pavel Surynek (v 1,v 2 ) (v 1,v 1 ) (v 1,v n ) (v 2,v 2 ) (v 2,v 1 ) (v 2,v n ) (v n,v 2 ) (v n,v 1 ) (v n,v n )... (v i,v k ) (v j,v k+1 ) {v i,v j }  E L1L1 L2L2 LnLn

Approximation Algorithm We need to relax from the exact enforcing the modified path-consistency We use a variant of Dijkstra’s single-source shortest path algorithm  for each vertex we calculate the lower bound of number of visits in layers  if the lower bound of visits in L i for the vertex x is k then every path starting in the original vertex and ending in x visits L i at least k-times  lower bounds allow us to check violation of the maximum number of visits → some paths may become forbidden CSCLP 2008 Pavel Surynek

Experimental Evaluation: propagation strength of PC and MPC Comparison of the number of filtered pairs of values  several benchmark problems from the SAT Library  comparison of PC and modified PC enforced by approximation algorithm  on some problems modified PC is significantly stronger  times were slightly higher for modified PC CSCLP 2008 Pavel Surynek SAT Problem Number of variables Number of clauses Pairs filtered by standard PC Pairs filtered by modified PC bw_large.a hanoi huge jnh logistics.a medium par8-1-c par8-2-c par8-3-c par16-1-c par16-2-c par16-3-c ssa0432/ ssa2670/ ssa2670/ ssa7552/ ssa7552/

Experimental Evaluation: PC and MPC on SAT preprocessing Improvement ratio gained by preprocessing of SAT problems by modified PC in comparison with PC  the number of decision steps was measured  on some problems modified PC has a good effect CSCLP 2008 Pavel Surynek Problem#variables#clausesHaifaSatMinisat2Rsat_1_03zChaff bw_large.a hanoi hanoi huge jnh logistics.a medium par8-1-c par8-2-c par8-3-c par16-1-c par16-2-c par16-3-c ssa ssa ssa ssa ssa

Conclusions and Future Work We proposed a modified variant of path-consistency We are trying to exploit global structural properties of the problem Experimental evaluation indicates some advantages of modified path-consistency in comparison with the standard version There are still many open questions for future work:  experimental evaluation on standard CSPs  more competitive evaluation with SAT solvers  better approximation algorithm, tractable cases CSCLP 2008 Pavel Surynek