Integrating symmetry breaking into a DLL procedure

Many real-world problems have interesting symmetries. Encoding these problems into CNF formulas generally results in hard SAT problems. Examples includes the pigeon hole problem, which, although very simple, is nonpolynomial for any resolution-based method when encoded into SAT [6], and the n queen problem. In order to speed up search algorithms for these problems, intrinsic symmetries should be exploited to avoid repeated search of equivalent portions of search space. The general strategy to exploit symmetries is to divide the objects of the search space into equivalence classes using symmetries, so that only one object in each class needs to be examined by a search algorithm. Brown, Finkelstein and Purdom [3] integrated symmetrydetection into a general-purpose backtracking search with dynamic variable search order, making intensive use of results from computational group theory. In the context of propositional reasoning, Benhamou and Sais [2] detected syntactic symmetries in a CNF formula and used these symmetries to reduce the search space. Crawford, Ginsberg and Luks [4] proposed a general schema to add symmetry-breaking axioms as CNF clauses into a CNF formula, which can be used as a pre-processor to any propositional reasoning method. In general case adding symmetry-breaking axioms appears to be intractable. Crawford, Ginsberg and Luks limited the number of added axioms to be polynomial. Shlyakhter [12] extended the approach of Crawford, Ginsberg and Luks by describing polynomial-size symmetry-breaking predicates for some common combinatorial objects. In this paper we propose a generic symmetry-breaking schema in a DLL procedure [5], inspired from the approach of Brown, Finkelstein and Purdom. Like their approach, we exploit symmetries in every node of a search tree so that only one object in every equivalence class is searched by the DLL procedure. Unlike their approach, we are in propositional reasoning case and don’t use results of computational group theory. We present our approach using two This work is supported by French CNRS under grant number SUB/2001/0111/DR16