The Complexity of General-Valued CSPs

TL;DR

This paper investigates the computational complexity of the Valued Constraint Satisfaction Problem (VCSP), establishing conditions under which VCSPs are tractable and linking their complexity to that of classical CSPs.

Contribution

It proves that if a constraint language meets an algebraic necessary condition and the corresponding feasibility problem is tractable, then the VCSP is also tractable, connecting VCSP complexity to classical CSP dichotomy.

Findings

Main result links VCSP tractability to feasibility CSP tractability under algebraic conditions.

Proposes a simple algorithm combining feasibility algorithms and LP relaxation.

Shows that a dichotomy for CSPs implies a dichotomy for VCSPs.

Abstract

An instance of the Valued Constraint Satisfaction Problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite values specifying costs of assignments of labels to its variables or the infinite value, which indicates infeasible assignments. The goal is to find an assignment of labels to the variables that minimizes the sum. We study (assuming that P $\ne$ NP) how the complexity of this very general problem depends on the set of functions allowed in the instances, the so-called constraint language. The case when all allowed functions take values in $\{0,\infty\}$ corresponds to ordinary CSPs, where one deals only with the feasibility issue and there is no optimization. This case is the subject of the Algebraic CSP Dichotomy Conjecture predicting for which constraint languages CSPs are tractable and for which NP-hard. The case when all allowed functions take only finite values corresponds to finite-valued CSP, where the feasibility aspect is trivial and one deals only with the optimization issue. The complexity of finite-valued CSPs was fully classified by Thapper and \v{Z}ivn\'y. An algebraic necessary condition for tractability of a general-valued CSP with a fixed constraint language was recently given by Kozik and Ochremiak. As our main result, we prove that if a constraint language satisfies this algebraic necessary condition, and the feasibility CSP corresponding to the VCSP with this language is tractable, then the VCSP is tractable. The algorithm is a simple combination of the assumed algorithm for the feasibility CSP and the standard LP relaxation. As a corollary, we obtain that a dichotomy for ordinary CSPs would imply a dichotomy for general-valued CSPs.