The complexity of finite-valued CSPs

TL;DR

This paper establishes a clear dichotomy in the computational complexity of exactly solving finite-valued constraint satisfaction problems, showing they are either efficiently solvable by linear programming or as hard as Max-Cut.

Contribution

It proves a dichotomy theorem classifying all finite-valued VCSPs as either tractable via LP or NP-hard, depending on their algebraic properties.

Findings

LP relaxation solves all instances with certain polymorphisms

Some languages are NP-hard via reduction from Max-Cut

Dichotomy applies to all finite domains

Abstract

We study the computational complexity of exact minimisation of rational-valued discrete functions. Let $\Gamma$ be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, $\operatorname{VCSP}(\Gamma)$, is the problem of minimising a function given as a sum of functions from $\Gamma$. We establish a dichotomy theorem with respect to exact solvability for all finite-valued constraint languages defined on domains of arbitrary finite size. We show that every constraint language $\Gamma$ either admits a binary symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of $\operatorname{VCSP}(\Gamma)$ exactly, or $\Gamma$ satisfies a simple hardness condition that allows for a polynomial-time reduction from Max-Cut to $\operatorname{VCSP}(\Gamma)$.