Abstract
AbstractWe determine the complexity of approximate counting of the total weight of assignments for complex-weighted Boolean constraint satisfaction problems (or CSPs), particularly, when degrees of instances are bounded from above by a given constant, provided that all arity-1 (or unary) constraints are freely available. All degree-1 counting CSPs are solvable in polynomial time. When the degree is more than 2, we present a trichotomy theorem that classifies all bounded-degree counting CSPs into only three categories. This classification extends to complex-weighted problems an earlier result on the complexity of the approximate counting of bounded-degree unweighted Boolean CSPs. The framework of the proof of our trichotomy theorem is based on Cai’s theory of signatures used for holographic algorithms. For the degree-2 problems, we show that they are as hard to approximate as complex Holant problems.KeywordsBipartite GraphUnary ConstraintBoolean VariableUnary SignatureBinary SignatureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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