Abstract
The solution graph of a Boolean formula on n variables is the subgraph of the hypercube \(H_n\) induced by the satisfying assignments of the formula. The structure of solution graphs has been the object of much research in recent years since it is important for the performance of SAT-solving procedures based on local search. Several authors have studied connectivity problems in such graphs focusing on how the structure of the original formula might affect the complexity of the connectivity problems in the solution graph.
In this paper we study the complexity of the isomorphism problem of solution graphs of Boolean formulas and we investigate how this complexity depends on the formula type.
We observe that for general formulas the solution graph isomorphism problem can be solved in exponential time while in the cases of 2CNF formulas, as well as for CPSS formulas, the problem is in the counting complexity class \(\text {C}_=\text {P} \), a subclass of PSPACE. We also prove a strong property on the structure of solution graphs of Horn formulas showing that they are just unions of partial cubes.
In addition we give a \(\text {PSPACE} \) lower bound for the problem on general Boolean functions. We prove that for 2CNF, as well as for CPSS formulas the solution graph isomorphism problem is hard for \(\text {C}_=\text {P} \) under polynomial time many one reductions, thus matching the given upper bound.
P. Scharpfenecker—Supported by DFG grant TO 200/3-1.
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Notes
- 1.
Identifying two variables corresponds to replacing one of them with the other variable.
- 2.
Boolean isomorphisms are signed permutations: they may map variables to variables and may flip variables.
- 3.
We use the same quantifier notation which is common for the classes in the polynomial time hierarchy.
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Scharpfenecker, P., Torán, J. (2016). Solution-Graphs of Boolean Formulas and Isomorphism. In: Creignou, N., Le Berre, D. (eds) Theory and Applications of Satisfiability Testing – SAT 2016. SAT 2016. Lecture Notes in Computer Science(), vol 9710. Springer, Cham. https://doi.org/10.1007/978-3-319-40970-2_3
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