Approximate Counting for Complex-Weighted Boolean Constraint Satisfaction Problems

Abstract

Constraint satisfaction problems (or CSPs) have been extensively studied in, e.g.,artificial intelligence, database theory, graph theory, and statistical physics. A practical application often requires only an approximate value of the total number of assignments that satisfy all given Boolean constraints. There is a known trichotomy theorem for such approximate counting for (non-weighted) Boolean CSPs; namely, all such counting problems are neatly classified into three categories under polynomial-time approximation-preserving reductions. We extend this result to approximate counting for complex-weighted Boolean CSPs, provided that all unary constraints are freely available to use. This marks a significant progress in the quest for the approximation classification of all counting Boolean CSPs. To deal with complex weights, we employ proof techniques along the line of solving Holant problems. Our result also gives an approximation version of the known dichotomy theorem of the complexity of exact counting for such complex-weighted Boolean CSPs.