 
Summary: Efficient Symmetry Breaking for
Boolean Satisfiability
Fadi A. Aloul, Member, IEEE, Karem A. Sakallah, Fellow, IEEE, and
Igor L. Markov, Senior Member, IEEE
AbstractIdentifying and breaking the symmetries of conjunctive normal form (CNF) formulae has been shown to lead to significant
reductions in search times. Symmetries in the search space are broken by adding appropriate symmetrybreaking predicates (SBPs)
to an SAT instance in CNF. The SBPs prune the search space by acting as a filter that confines the search to nonsymmetric regions of
the space without affecting the satisfiability of the CNF formula. For symmetry breaking to be effective in practice, the computational
overhead of generating and manipulating SBPs must be significantly less than the runtime savings they yield due to search space
pruning. In this paper, we describe a more systematic and efficient construction of SBPs. In particular, we use the cycle structure of
symmetry generators, which typically involve very few variables, to drastically reduce the size of SBPs. Furthermore, our new SBP
construction grows linearly with the number of relevant variables as opposed to the previous quadratic constructions. Our empirical
data suggest that these improvements reduce search runtimes by one to two orders of magnitude on a wide variety of benchmarks with
symmetries.
Index TermsBacktrack Search, clause learning, conjunctive normal form (CNF), graph automorphism, satisfiability (SAT),
symmetries.
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1 INTRODUCTION
MODERN Boolean satisfiability (SAT) solvers, based on
backtrack search, are now capable of attacking
