Complexity of generalized satisfiability counting problems
Information and Computation
On the Structure of Polynomial Time Reducibility
Journal of the ACM (JACM)
Information Processing Letters
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Quantum Circuits That Can Be Simulated Classically in Polynomial Time
SIAM Journal on Computing
Theoretical Computer Science
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Inapproximability of the Tutte polynomial
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
SIAM Journal on Computing
Signature Theory in Holographic Algorithms
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Holant problems and counting CSP
Proceedings of the forty-first annual ACM symposium on Theory of computing
The Complexity of Weighted Boolean CSP
SIAM Journal on Computing
An approximation trichotomy for Boolean #CSP
Journal of Computer and System Sciences
Holographic algorithms: From art to science
Journal of Computer and System Sciences
Approximate counting for complex-weighted Boolean constraint satisfaction problems
WAOA'10 Proceedings of the 8th international conference on Approximation and online algorithms
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Constraint satisfaction problems (or CSPs) have been extensively studied in, for instance, artificial intelligence, database theory, graph theory, and statistical physics. From a practical viewpoint, it is beneficial to approximately solve those CSPs. When one tries to approximate the total number of truth assignments that satisfy all Boolean-valued constraints for (unweighted) Boolean CSPs, there is a known trichotomy theorem by which all such counting problems are neatly classified into exactly three categories under polynomial-time (randomized) approximation-preserving reductions. In contrast, we obtain a dichotomy theorem of approximate counting for complex-weighted Boolean CSPs, provided that all complex-valued unary constraints are freely available to use. It is the expressive power of free unary constraints that enables us to prove such a stronger, complete classification theorem. This discovery makes a step forward in the quest for the approximation-complexity classification of all counting CSPs. To deal with complex weights, we employ proof techniques of factorization and arity reduction along the line of solving Holant problems. Moreover, we introduce a novel notion of T-constructibility that naturally induces approximation-preserving reducibility. Our result also gives an approximation analogue of the dichotomy theorem on the complexity of exact counting for complex-weighted Boolean CSPs.