Balance and Filtering in Structured Satisfiable Problems

New methods to generate hard random problem instances have driven progress on algorithms for deduction and constraint satisfaction. Recently Achlioptas et al. (AAAI 2000) introduced a new generator based on Latin squares that creates only satisfiable problems, and so can be used to accurately test incomplete (one sided) solvers. We investigate how this and other generators are biased away from the uniform distribution of satisfiable problems and show how they can be improved by imposing a balance condition. More generally, we show that the generator is one member of a family of related models that generate distributions ranging from ones that are everywhere tractable to ones that exhibit a sharp hardness threshold. We also discuss the critical role of the problem encoding in the performance of both systematic and local search solvers.

[1]  Tad Hogg,et al.  Phase Transitions and the Search Problem , 1996, Artif. Intell..

[2]  W. Vega.  On the Bandwidth of Random Graphs , 1983 .

[3]  William G. Poole,et al.  An algorithm for reducing the bandwidth and profile of a sparse matrix , 1976 .

[4]  ZhangHantao,et al.  Implementing the DavisPutnam Method , 2000 .

[5]  David G. Mitchell,et al.  Finding hard instances of the satisfiability problem: A survey , 1996, Satisfiability Problem: Theory and Applications.

[6]  M. Stickel,et al.  Automated reasoning and exhaustive search: Quasigroup existence problems☆ , 1995 .

[7]  Bart Selman,et al.  Problem Structure in the Presence of Perturbations , 1997, AAAI/IAAI.

[8]  M. Hall.  An existence theorem for latin squares , 1945 .

[9]  Hector J. Levesque,et al.  A New Method for Solving Hard Satisfiability Problems , 1992, AAAI.

[10]  Chu Min Li,et al.  Heuristics Based on Unit Propagation for Satisfiability Problems , 1997, IJCAI.

[11]  Roberto J. Bayardo,et al.  Using CSP Look-Back Techniques to Solve Exceptionally Hard SAT Instances , 1996, CP.

[12]  Toby Walsh,et al.  The Difference All-Difference Makes , 1999, IJCAI.

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

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

[15]  Peter C. Cheeseman,et al.  Where the Really Hard Problems Are , 1991, IJCAI.

[16]  Jean-Charles Régin,et al.  A Filtering Algorithm for Constraints of Difference in CSPs , 1994, AAAI.

[17]  Toby Walsh,et al.  An Empirical Analysis of Search in GSAT , 1993, J. Artif. Intell. Res..

[18]  Bart Selman,et al.  Generating Satisfiable Problem Instances , 2000, AAAI/IAAI.

[19]  H. Ryser.  A combinatorial theorem with an application to latin rectangles , 1951 .

[20]  Hantao Zhang,et al.  SATO: An Efficient Propositional Prover , 1997, CADE.

[21]  Leonid A. Levin,et al.  Pseudo-random generation from one-way functions , 1989, STOC '89.