The Complexity of Symmetry-Breaking Formulas

Symmetry-breaking formulas for a constraint-satisfaction problem are satisfied by exactly one member (e.g., the lexicographic leader) from each set of “symmetrical points” in the search space. Thus, the incorporation of such formulas can accelerate the search for a solution without sacrificing satisfiability. We study the computational complexity of generating lex-leader formulas. We show, even for abelian symmetry groups, that the number of essential clauses in the “natural” lex-leader formula could be exponential. Furthermore, we show the intractability (NP-hardness) of finding any expression of lex-leadership without reordering the variables, even for elementary abelian groups with orbits of size 3. Nevertheless, using techniques of computational group theory, we describe a reordering relative to which we construct small lex-leader formulas for abelian groups.

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