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

Constraint satisfaction problems have been studied in numerous fields with practical and theoretical interests. In recent years, major breakthroughs have been made in a study of counting constraint satisfaction problems (or #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 input instance is the maximal number of times that each input variable appears in a given set of constraints. Despite the efforts of recent studies, however, a complexity classification of degree-2 #CSPs has eluded from our understandings. This paper challenges this open problem and gives its partial solution by applying two novel proof techniques--T_{2}-constructibility and parametrized symmetrization--which are specifically designed to handle "arbitrary" constraints under randomized approximation-preserving reductions. We partition entire constraints into four sets and we classify the approximation complexity of all degree-2 #CSPs whose constraints are drawn from two of the four sets into two categories: problems computable in polynomial-time or problems that are at least as hard as #SAT. Our proof exploits a close relationship between complex-weighted degree-2 #CSPs and Holant problems, which are a natural generalization of complex-weighted #CSPs.