MiniMaxSat: : A new Weighted Solver. Federico Heras Javier Larrosa Albert Oliveras

Transcription

1 MiniMaxSat: : A new Weighted Max-SAT Solver Federico Heras Javier Larrosa Albert Oliveras

2 SAT (x v y), ( y v z), ( z v w)

3 SAT (x v y), ( y v z), ( z v w) x=y=z=w=true Satisfiable

4 SAT (x), ( x v y), ( x v y v z), ( x v y v z)

5 SAT (x), ( x v y), ( x v y v z), ( x v y v z) Unsatisfiable

6 Max-SAT (x), ( x v y), ( x v y v z), ( x v y v z)

7 Max-SAT (x), ( x v y), ( x v y v z), ( x v y v z) x=y=z=true Optimum: 1

8 Weighted Max-SAT (x,5), ( x v y,6), ( x v y v z,3), ( x v y v z,7)

9 Weighted Max-SAT (x,5), ( x v y,6), ( x v y v z,3), ( x v y v z,7) x=y=true, z=false Optimum: 3

10 Overview (Weighted) Max-SAT: Given a set of weighted clauses, find an assignment of the variables such that the sum of the weights of the violated clauses is minimized. Complexity: NP-Hard Hard.

11 Overview Applications: Max-Cut Max-Clique Bayesian Networks Combinatorial Auctions Binate Covering Problem... Recent Algorithms: Focused on computing better lower boundings for Max-SAT.

12 MiniMaxSat: MiniSat: Contributions Two Watched literals. Learning and backjumping. Novel: Lower bound based on the best previous approaches. Extension of probing to Max-SAT. Pseudo boolean optimization to Max-SAT.

13 Outline Preliminaries Search Lower Bound Additional Features Empirical Results Conclusions and Future Work

14 Max-SAT Notation (C,T) (l,t) (,lb) ub Hard Clauses, otherwise soft clauses Unit propagation over hard unit clauses Lower Bound Upper Bound Hard Conflict: A hard clause is falsified. Soft Conflict: lb >= ub

15 Outline Preliminaries Search Lower Bound Additional Features Empirical Results Conclusions and Future Work

16 Search Function Search() ub:= :=LocalSearch(); Loop Propagate(); if (Hard Conflict) LearnAndBackjump (); else if (Soft Conflict) ChronoBactrack (); else if (Complete Assignment) ub=lb lb; ChronoBactrack (); else SelectLit();

17 Outline Preliminaries Search Lower Bound Additional Features Empirical Results Conclusions and Future Work

18 Previous work Inconsistent subset: Subset of clauses that cannot be satisfied simultaneously by any assignment. Hyper-resolution[Heras06]. UP lower bound [Li05].

19 Weighted Resolution (Max-RES) (x A,u), ( x B,w) (where m=min{u,w}) (A B,m), (x A,u-m), ( x B, w-m), (x A B,m), ( x A B,m) [Larrosa IJCAI 05] Complete calculus: : [Bonet[ SAT 06]

20 Weighted Resolution (Max-RES) Clashing clauses (x A,u), ( x B,w) (where m=min{u,w}) (A B,m), (x A,u-m), ( x B, w-m), (x A B,m), ( x A B,m) Resolvent Posterior clashing clauses Compensation clauses [Larrosa IJCAI 05] Complete calculus: : [Bonet[ SAT 06]

21 A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) Lower bound

22 A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) Lower bound x(a)

23 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(d) y(c) w(b) x(a)

24 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(d) y(c) w(b) x(a)

25 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(e) z(d) y(c) w(b) x(a)

26 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(e) z(d) y(c) w(b) x(a)

27 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) Conflicting clause z(e) z(d) y(c) w(b) x(a)

28 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(e) z(d) y(c) w(b) x(a)

29 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(e) z(d) y(c) w(b) x(a) E

30 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(e) z(d) E x v y D y(c) w(b) x(a)

31 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(e) z(d) E x v y D C y(c) w(b) x x(a)

32 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(e) z(d) E x v y D C y(c) w(b) x A x(a)

33 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(e) z(d) ( y v z,3) (x v y,2) (x v z,2) (x v y,t) y(c) w(b) (x,2) ( x,2) x(a) (,2) ( x v z,1), ( x v y v z,2), (x v y v z,2), (x v y,t), (x v w,1)

34 Lower bound A = ( x,2) B = (x v w,1) C = (x v y,t) D = (x v z,2) E = ( y v z,3) z(e) z(d) ( y v z,3) (x v y,2) (x v z,2) (x v y,t) Repeat the process until UP does not lead to falsify a clause y(c) w(b) x(a) (x,2) (,2) ( x,2) ( x v z,1), ( x v y v z,2), (x v y v z,2), (x v y,t), (x v w,1)

35 Patterns with non-binary clauses ( x,w 1 ), ( y, w 2 ), ( z, w 3 ), (x v y v z, w 4 ) ( x, w 1 ), (x v y, w 2 ), ( z, w 3 ), (x v y v z, w 4 ) ( x, w 1 ), (x v y, w 2 ), (x v y v z, w 3 ), (q v z, w 4 ), ( q, w 5 )

36 Outline Preliminaries Search Lower Bound Additional Features Empirical Results Conclusions and Future Work

37 Probing Add a unit clause (l,t). Simulate unit propagation. Build reslution tree. Apply Max-Sat resolution. As a result, add a new unit clause ( l,w). A = ( x v y,1) B = ( x v z,1) C = ( y v z,1) C ( x,1) B A

38 Pseudo Boolean to Max-SAT PBO: Minimization function (c 1 x 1 +c 2 x 2 + +c n x n ). Pseudo boolean constraints. To Max-SAT: Min. Function: For each term c i x i create a new unit soft clause ( x i,c i ). Pseudo boolean constraints: To hard clauses using BDDs, Sorters or Adders [Een06].

39 Branching heuristic Depending on the problem structure: Compute Jeroslow-like heuristic in the root node. Compute VSIDS-like heuristic on hard conflicts.

40 Outline Preliminaries Max-SAT Search Resolution for Max-SAT Lower Bound Additional Features Empirical Results Conclusions and Future Work

41 Results Solvers from different communities: Maxsatz: Unweighted Max-SAT solver. Max-DPLL (Toolbar): Weighted Max-SAT solver. Toolbar: WCSP solver. Pueblo: Pure pseudo-boolean optimizer. Minisat+: Translates to several instances of SAT. Pentium 4, 3 Ghz, Linux, 10 mins limit.

42 Results (Max-SAT, Max-CUT) MaxSatz MiniMaxSat Max-DPLL

43 Results (Max-Clique,Max-One) MaxClique DIMACS: 34, second Max-DPLL 29. MiniMaxSat Max-DPLL Pueblo Minisat+

44 Results (Comb. Auctions) MiniMaxSat Max-DPLL Pueblo Minisat+

45 Results WCSP direct encoding: Planning and Max-CSP PBE: Garden, Min Prime, Logic Synthesis, Miplib

46 Outline Preliminaries Max-SAT Search Resolution for Max-SAT Lower Bound Additional Features Empirical Results Conclusions and Future Work

47 Conclusions MiniMaxSat: Novel lower bound and probing. Integrates SAT and Max-SAT techniques. PBO to Max-SAT. Very robust Weighted Max-SAT solver.

48 Future Work Adapt VSIDS heuristic to soft clauses. Incorporate Backjumping over soft conflicts.