Symmetry-Breaking Predicates for Search Problems

Many reasoning and optimization problems exhibit symmetries. Previous work has shown how special purpose algorithms can make use of these symmetries to simplify reasoning. We present a general scheme whereby symmetries are exploited by adding \symmetry-breaking" predicates to the theory. Our approach can be used on any propo-sitional satissability problem, and can be used as a pre-processor to any (systematic or non-systematic) reasoning method. In the general case adding symmetry-breaking axioms appears to be intractable. We discuss methods for generating partial symmetry-breaking predicates, and show that in several speciic cases symmetries can be broken either fully are partially using a polynomial number of predicates. These ideas have been implemented and we include experimental results on two classes of constraint-satisfaction problems.

[1]  Eugene C. Freuder.  Eliminating Interchangeable Values in Constraint Satisfaction Problems , 1991, AAAI.

[2]  Steven Minton,et al.  Solving Large-Scale Constraint-Satisfaction and Scheduling Problems Using a Heuristic Repair Method , 1990, AAAI.

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

[4]  László Babai,et al.  Canonical labelling of graphs in linear average time , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[5]  Eugene M. Luks,et al.  Isomorphism of graphs of bounded valence can be tested in polynomial time , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[6]  Paul Walton Purdom,et al.  Backtrack Searching in the Presence of Symmetry , 1988, Nord. J. Comput..

[7]  László Babai,et al.  Canonical labeling of graphs , 1983, STOC.

[8]  Lakhdar Sais,et al.  Theoretical Study of Symmetries in Propositional Calculus and Applications , 1992, CADE.

[9]  Takunari Miyazaki,et al.  The complexity of McKay's canonical labeling algorithm , 1995, Groups and Computation.

[10]  Gregory Butler,et al.  A General Backtrack Algorithm for the Isomorphism Problem of Combinatorial Objects , 1985, J. Symb. Comput..

[11]  Matthew L. Ginsberg,et al.  Dynamic Backtracking , 1993, J. Artif. Intell. Res..

[12]  Clement W. H. Lam.  Application of Group Theory to Combinatorial Searches , 1991, Groups And Computation.

[13]  Clement W. H. Lam,et al.  Backtrack Search with Isomorph Rejection and Consistency Check , 1989, J. Symb. Comput..

[14]  J. Gaschnig.  Performance measurement and analysis of certain search algorithms. , 1979 .

[15]  James Mackenzie Crawford.  A theoretical analysis of reasoning by symmetry in first-order logic (extended abstract) , 1992 .

[16]  B. McKay.  nauty User ’ s Guide ( Version 2 . 4 ) , 1990 .