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

TL;DR

This paper advances the understanding of the computational complexity of degree-2 counting constraint satisfaction problems (#CSPs) with complex weights by introducing novel proof techniques and classifying problems into polynomial-time solvable or #SAT-hard categories.

Contribution

It provides a partial complexity classification for degree-2 #CSPs using new proof methods, addressing a longstanding open problem.

Findings

Classified degree-2 #CSPs into polynomial-time or #SAT-hard categories.

Introduced T_{2}-constructibility and parametrized symmetrization techniques.

Linked complex-weighted degree-2 #CSPs to Holant problems.

Abstract

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.