Combination of Exact and Approximate Methods for SAT and MAX-SAT Problems

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

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

3. 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

4. Searchtree a F T b b F T F V c c c c T F T T F F T T … … … … … … … … Introduction • Two classes of methods : - complete/exact methods (1960) • SAT : Davis Putnam, Satz (97), Zchaff (00), … • MAX-SAT : B&B

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

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

7. 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)

8. Selection Crossover Local Search Insertion Result GASAT Algorithm Conditions

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

10. 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]

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

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

13. a F V b b F V F V c c c c V F V F F F V V … … … … … … … … Existing hybridizations Initial bound given by GSAT, then B&B (Borchers et Furman 99)

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

15. Tri-valued Tabusearch to unify the two 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 Difficulties : complete assignments (Tabu) v.s. partial assignments (B&B) Proposition : a Tabou search with 3 truth values: true, false et undefined

16. Tri-valued Tabu Search • 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

17. 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

18. 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

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

20. Hybridization Tabu + B&B

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

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

23. Experimental results 1 000 000 of flips (1 backtrack = 1 flip)

24. 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

25. 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…)

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