Combination of Exact and Approximate Methods for SAT and MAX-SAT Problems Frédéric Lardeux, Frédéric Saubion and Jin-Kao Hao Metaheuristics and Combinatorial Optimization Group LERIA, University of Angers (France)

SAT Problem : NP-complete problem, several applications (planning, diagnosis …) Goal : Find a truth assignment satisfying a propositional logic formula: Introduction Ex : (  a  b  c)  (  b  d  e)  (a   c  f) TTFTFF fedcba

Introduction MAX-SAT problem: Goal : For a given boolean formula, find an assignment which maximizes the number of true clauses (or minimizes the number of false clauses) Remark : MAXSAT is less studied than SAT

Two classes of methods : - complete/exact methods (1960) Introduction Search tree a bb c cc …………… F F F F F T T T T V T T … c … FT … SAT : Davis Putnam, Satz (97), Zchaff (00), … MAX-SAT : B&B

Two classes of methods : - heuristic methods (1992) GSAT(Walksat), UnitWalk(2002), … Search space Set of assignments Introduction

Outline Hybrid Solvers for SAT New hybridization scheme Tri-valued Tabu Search Hybridization TS + B&B Experimental results Conclusions

Hybrid Solvers for SAT Hybridization/ Combination = popular technique for designing powerful algorithms Our experience : GASAT : hybrid incomplete SAT solver evolutionnary process (population + recombination) + local search (tabu)

Local Search GASAT Algorithm Insertion Crossover Selection Result Conditions

GASAT Algorithm Interesting Results SAT 2004 : 4 th for Random Benchmarks Looking for new hybridizations complete + incomplete Focusing on MAXSAT

Existing hybridizations complete/incomplete 3 hybrid algorithms : Tabu Search process in a DP procedure used as a branching rule (Mazure et al. 98) [SAT] Walksat used in Satz to complete a partial assignment (Habet et al. 2002) [SAT] GSAT to obtain the initial bound, then B&B (Borchers et Furman 99) [MAXSAT]

Existing hybridizations DP with Tabu as branching rule (Mazure et al. 98) : a bb c cc …………… F F F F F F V V V V V V … c … FV … Given thanks to Tabu Davis Putnam+Tabu

Existing hybridizations Walksat completes a partial assignment (Habet et al. 2002) a bb c cc …………… F F F F F F V V V V V V … c … FV … Satz and Walksat Walksat

Existing hybridizations Initial bound given by GSAT, then B&B (Borchers et Furman 99) a bb c cc …………… F F F F F F V V V V V V … c … FV …

Outline Existing hybridizations New hybridization scheme Tri-valued Tabu Search Hybridization Tabu + B&B Experimental results Conclusions

Tri-valued Tabu Search Idea : use B&B in a Tabu Search process as an intensification mecanism: Tabu identifies « promising » areas B&B explores exhaustively these areas Tri-valued Tabu search to unify the two Proposition : a Tabou search with 3 truth values: true, false et undefined Difficulties : complete assignments (Tabu) v.s. partial assignments (B&B)

Main components of the tri-valued tabu Configuration with three values: true, false, undefined Fitness function: number of true clauses (or false) AND undefined (the aim is to maximize the number of true clauses) Neighborhood: change of one value in the current assignment Tri-valued Tabu Search

Rm : Each configuration corresponds to one or several complete assignments Ex: (T,F,U) = (T,F,F) and (T,F,T) The move from « undefined » to T or F corresponds to a (small) intensification The move from T or F to « undefined » corresponds to a diversification Tri-valued Tabu Search

S: standard move: T to F or F to T I: intensification move (small): Undefined to T or F D: diversification move: T or F to Undefined

Hybridization Tabu + B&B 1.(Intensification/Diversification) tri-valued tabu search until the obtention of an assignment with at the maximum k undefined variables 2.(Intensification) B&B on the sub-problem of k variables to value them. Obtention of a complete assignment s 3.(Update) Add the k variables to the tabu list 4.Return to 1 with the solution s

Hybridization Tabu + B&B

Outline  Existing hybridizations  New hybridization  Tri-valued Tabu Search  Hybridization Tabu + B&B  Experimental results  Conclusions

Experimental results Evaluate the hybrid algorithm w.r.t. a standard Tabu Search Benchmarks: SAT 2003 competition

Experimental results BenchmarksTabuHybridization (Tri-valued TS + B&B) Imp. instancesvarclsf.c.s.d.f.c.s.d.% par32-5-c par Mat25.shuffled Mat26.shuffled f f of flips (1 backtrack = 1 flip)

Conclusions Preliminary study of a hybridization between Tabu and B&B with a new resolution work Improvement of the standard tabu search Worst results than the best SAT solvers

Future Works General theoretical framework for hybrid CSP solvers Based on K.R. Apt’s chaotic iteration LS + CP [ICLP2004 with E. Monfroy and T. Lambert] Study of the parameters (balancing between the two stages of the algorithm), Hybridization of other methods with the same resolution framework (Walksat + B&B…)

Combination of Exact and Approximate Methods for SAT and MAX-SAT Problems Frédéric Lardeux, Frédéric Saubion and Jin-Kao Hao University of Angers (France)