The Power of Linear Programming for Valued CSPs

TL;DR

This paper provides an algebraic characterization of valued constraint satisfaction problems (VCSPs) solvable exactly by linear programming, enabling new tractability results for classes like submodular and bisubmodular VCSPs.

Contribution

It offers a precise algebraic characterization of VCSPs solvable by linear programming, expanding understanding of problem classes with tractable solutions.

Findings

Characterization of VCSPs solvable by LP relaxation

Tractability results for submodular VCSPs on lattices

Tractability results for bisubmodular and tree-submodular VCSPs

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. This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs). Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation. Using this result, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) bisubmodular (also known as k-submodular) on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.