Approximation Complexity of Complex-Weighted Degree-Two Counting Constraint Satisfaction Problems

Abstract

Constraint satisfaction problems have been studied in numerous fields with practical and theoretical interests. Recently, major breakthroughs have been made in the study of counting constraint satisfaction problems (or simply #CSPs). In particular, a computational complexity classification of bounded-degree #CSPs has been discovered for all degrees except for two, where the degree of an instance is the maximal number of times that each input variable appears in any given set of constraints. This paper challenges an open problem of classifying all degree-2 #CSPs on an approximate counting model and presents its partial solution by developing two novel proof techniques—T₂-constructibility and parametrized symmetrization—which are specifically designed to handle arbitrary constraints under approximation-preserving reductions. Our proof exploits a close relationship between complex-weighted degree-2 #CSPs and Holant problems, which are a natural generalization of complex-weighted #CSPs.