An efficient local search method for random 3-satisfiability

Abstract We report on some exceptionally good results in the solution of randomly generated 3-satisfiability instances using the “record-to-record travel (RRT)” local search method. When this simple, but less-studied algorithm is applied to random one-million variable instances from the problem's satisfiable phase, it seems to find satisfying truth assignments almost always in linear time, with the coefficient of linearity depending on the ratio α of clauses to variables in the generated instances. RRT has a parameter for tuning “greediness”. By lessening greediness, the linear time phase can be extended up to very close to the satisfiability threshold α c . Such linear time complexity is typical for random-walk based local search methods for small values of α . Previously, however, it has been suspected that these methods necessarily lose their time linearity far below the satisfiability threshold. The only previously introduced algorithm reported to have nearly linear time complexity also close to the satisfiability threshold is the survey propagation (SP) algorithm. However, SP is not a local search method and is more complicated to implement than RRT. Comparative experiments with the WalkSAT local search algorithm show behavior somewhat similar to RRT, but with the linear time phase not extending quite as close to the satisfiability threshold.

[1]  Walter Kern,et al.  On the Depth of Combinatorial Optimization Problems , 1993, Discret. Appl. Math..

[2]  Michael E. Saks,et al.  On the complexity of unsatisfiability proofs for random k-CNF formulas , 1998, STOC '98.

[3]  Bart Selman,et al.  Local search strategies for satisfiability testing , 1993, Cliques, Coloring, and Satisfiability.

[4]  Jie Wang,et al.  Average-case computational complexity theory , 1998 .

[5]  J. K. Lenstra,et al.  Local Search in Combinatorial Optimisation. , 1997 .

[6]  Thomas Stützle,et al.  Systematic vs. Local Search for SAT , 1999, KI.

[7]  A. Selman,et al.  Complexity theory retrospective II , 1998 .

[8]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[9]  Olivier Dubois,et al.  Typical random 3-SAT formulae and the satisfiability threshold , 2000, SODA '00.

[10]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[11]  W. Freeman,et al.  Generalized Belief Propagation , 2000, NIPS.

[12]  James M. Crawford,et al.  Experimental Results on the Crossover Point in Random 3-SAT , 1996, Artif. Intell..

[13]  Hector J. Levesque,et al.  Hard and Easy Distributions of SAT Problems , 1992, AAAI.

[14]  G. Dueck.  New optimization heuristics , 1993 .

[15]  Riccardo Zecchina,et al.  Coloring random graphs , 2002, Physical review letters.

[16]  Lefteris M. Kirousis,et al.  The probabilistic analysis of a greedy satisfiability algorithm , 2002, Random Struct. Algorithms.

[17]  Endre Szemerédi,et al.  Many hard examples for resolution , 1988, JACM.

[18]  E. Friedgut,et al.  Sharp thresholds of graph properties, and the -sat problem , 1999 .

[19]  S Kirkpatrick,et al.  Critical Behavior in the Satisfiability of Random Boolean Expressions , 1994, Science.

[20]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[21]  Mats G. Nordahl,et al.  Relaxation in graph coloring and satisfiability problems , 1998, ArXiv.

[22]  Patric R. J. Osterg Aa Rd.  Computer search for small complete caps , 2000 .

[23]  U. Schöning.  A probabilistic algorithm for k-SAT and constraint satisfaction problems , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[24]  M. Mézard,et al.  Random K-satisfiability problem: from an analytic solution to an efficient algorithm. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  R. Monasson,et al.  Relaxation and metastability in a local search procedure for the random satisfiability problem. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Bart Selman,et al.  Accelerating Random Walks , 2002, CP.

[27]  M. Mézard,et al.  Analytic and Algorithmic Solution of Random Satisfiability Problems , 2002, Science.

[28]  Bart Selman,et al.  Evidence for Invariants in Local Search , 1997, AAAI/IAAI.

[29]  Christian M. Reidys,et al.  Combinatorial Landscapes , 2002, SIAM Rev..

[30]  C.H. Papadimitriou,et al.  On selecting a satisfying truth assignment , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[31]  Rémi Monasson,et al.  Determining computational complexity from characteristic ‘phase transitions’ , 1999, Nature.