The complexity of complex weighted Boolean #CSP

We prove a complexity dichotomy theorem for the most general form of Boolean #CSP where every constraint function takes values in the complex number field C. This generalizes a theorem by Dyer, Goldberg and Jerrum [11] where each constraint function takes non-negative values. We first give a non-trivial tractable class of Boolean #CSP which was inspired by holographic reductions. The tractability crucially depends on algebraic cancelations which are absent for non-negative numbers. We then completely characterize all the tractable Boolean #CSP with complex valued constraints and show that we have found all the tractable ones, and every remaining problem is #P-hard. We also improve our result by proving the same dichotomy theorem holds for Boolean #CSP with max degree 3 (every variable appears at most three times). The concept of Congruity and Semi-congruity provides a key insight and plays a decisive role in both the tractability and hardness proofs. We also introduce local holographic reductions as a technique in hardness proofs.