The power of linear programming for valued CSPs: a constructive characterization

TL;DR

This paper characterizes exactly which valued constraint satisfaction problems can be efficiently solved by linear programming, establishing a polynomial-time check for the key algebraic condition and linking it to computational complexity.

Contribution

It proves the equivalence between solvability by basic linear programming and the existence of a binary commutative fractional polymorphism, providing a practical polynomial-time criterion.

Findings

Linear programming solves exactly those VCSPs with a binary commutative fractional polymorphism.

Languages not satisfying the condition are NP-hard, establishing a clear complexity dichotomy.

The new condition simplifies previous infinite inequalities to a polynomial-time check.

Abstract

A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. We study which classes of finite-valued languages can be solved exactly by the basic linear programming relaxation (BLP). Thapper and Zivny showed [20] that if BLP solves the language then the language admits a binary commutative fractional polymorphism. We prove that the converse is also true. This leads to a necessary and a sufficient condition which can be checked in polynomial time for a given language. In contrast, the previous necessary and sufficient condition due to [20] involved infinitely many inequalities. More recently, Thapper and Zivny [21] showed (using, in particular, a technique introduced in this paper) that core languages that do not satisfy our condition are NP-hard. Taken together, these results imply that a finite-valued language can either be solved using Linear Programming or is NP-hard.