Complexity of Counting CSP with Complex Weights

We give a complexity dichotomy theorem for the counting constraint satisfaction problem (#CSP in short) with algebraic complex weights. To this end, we give three conditions for its tractability. Let F be any finite set of algebraic complex-valued functions defined on an arbitrary finite domain. We show that #CSP(F) is solvable in polynomial time if all three conditions are satisfied and is #P-hard otherwise. Our dichotomy theorem generalizes a long series of important results on counting problems and reaches a natural culmination: (a) the problem of counting graph homomorphisms is the special case when F has a single symmetric binary function [Dyer and Greenhill 2000; Bulatov and Grohe 2005; Goldberg et al. 2010; Cai et al. 2013]; (b) the problem of counting directed graph homomorphisms is the special case when F has a single but not necessarily symmetric binary function [Dyer et al. 2007; Cai and Chen 2010]; (c) the unweighted form of #CSP is when all functions in F take values in {0, 1} [Bulatov 2008; Dyer and Richerby 2013].