Complexity of Counting CSP with Complex Weights

TL;DR

This paper establishes a comprehensive complexity classification for counting constraint satisfaction problems with complex weights, identifying precise conditions under which these problems are efficiently solvable or computationally hard.

Contribution

It introduces three tractability conditions for #CSP with complex weights and proves a dichotomy theorem generalizing previous counting problem results.

Findings

#CSP with complex weights is polynomial-time solvable if all three conditions are met.

Otherwise, #CSP with complex weights is #P-hard.

The results unify and extend known counting problem classifications.

Abstract

We give a complexity dichotomy theorem for the counting Constraint Satisfaction Problem (#CSP in short) with complex weights. To this end, we give three conditions for its tractability. Let F be any finite set of complex-valued functions, then we prove that #CSP(F) is solvable in polynomial time if all three conditions are satisfied; and is #P-hard otherwise. Our complexity dichotomy generalizes a long series of important results on counting problems: (a) the problem of counting graph homomorphisms is the special case when there is a single symmetric binary function in F; (b) the problem of counting directed graph homomorphisms is the special case when there is a single not-necessarily-symmetric binary function in F; and (c) the standard form of #CSP is when all functions in F take values in {0,1}.