Improved Static Symmetry Breaking for SAT

Abstract

An effective SAT preprocessing technique is the construction of symmetry breaking formulas: auxiliary clauses that guide a SAT solver away from needless exploration of symmetric subproblems. However, during the past decade, state-of-the-art SAT solvers rarely incorporated symmetry breaking. This suggests that the reduction of the search space does not outweigh the overhead incurred by detecting symmetry and constructing symmetry breaking formulas. We present three methods to construct more effective symmetry breaking formulas. The first method simply improves the encoding of symmetry breaking formulas. The second detects special symmetry subgroups, for which complete symmetry breaking formulas exist. The third infers binary symmetry breaking clauses for a symmetry group as a whole rather than longer clauses for individual symmetries. We implement these methods as a symmetry breaking preprocessor, and verify their effectiveness on both hand-picked problems as well as the 2014 SAT competition benchmark set. Our experiments indicate that our symmetry breaking preprocessor improves the current state-of-the-art in static symmetry breaking for SAT and has a sufficiently low overhead to improve the performance of modern SAT solvers on hard combinatorial instances.

CDCLSym: Introducing Effective Symmetry Breaking in SAT Solving

“…-MiniSAT, as the reference solver without symmetry handling [10]; -Shatter, a symmetry breaking preprocessor described in [2], coupled with the MiniSAT SAT engine; -breakID, another symmetry breaking preprocessor, described in [8], also coupled with the MiniSAT SAT engine.…”

“…These added predicates will prevent the solver from visiting equivalent (isomorphic) parts that eventually yield the same results [1,5]. This technique, called static symmetry breaking, has been implemented first in the state-of-the-art tool SHATTER [2] and then improved in BREAKID [8]. However, while giving excellent results on numerous symmetric problems, these approaches still fail to solve some classes of symmetric problems.…”

Symmetries of Quantified Boolean Formulas

“…A reasonable choice is to pick a set E of generators for G syn and let the inner conjunction run over (some of) the elements of E. The formula ψ of Thm. 21 can be efficiently encoded as conjunctive normal form (CNF), adopting the propositional encoding of [17,2]: let g ∈ G syn and let {y g 0 , . .…”

Beyond DRAT: Challenges in Certifying UNSAT

“…The presence of symmetries may cause an exponential blowup in the size of the SAT solver's search tree 2 , and a similar blowup in the size of the resulting unsatisfiability certificates. In order to break such symmetries, one can add so-called lex-leader constraints to the SAT instance that assert that applying a given symmetry to the assignment represented in the CNF does not produce a smaller assignment (see, for example, Devriendt et al [3]). The resulting formula is equisatisfiable to the original one, i.e., if there is a satisfying assignment, but a lex-leader constraint for a particular symmetry is violated, we may switch to the smaller satisfying assignment induced by the symmetry; this process must terminate.…”

“…As a minimal proposal, consider breaking syntactical symmetries 3 . In that case, the certificate contains a list of permutations, and list of literals defining a lexicographic order of assignments.…”