Sherali-Adams relaxations for valued CSPs

TL;DR

This paper investigates Sherali-Adams LP relaxations for valued CSPs, showing that many tractable cases have an integrality gap of 1 at a specific relaxation level, linking to bounded relational width.

Contribution

It extends the understanding of Sherali-Adams relaxations for valued CSPs, demonstrating their effectiveness in capturing tractability for a broad class of problems.

Findings

Most known tractable valued languages have an integrality gap of 1 for the (2,3)-Sherali-Adams relaxation.

The results relate to the concept of bounded relational width in classical CSPs.

Sherali-Adams relaxations effectively characterize tractability in valued CSPs.

Abstract

We consider Sherali-Adams linear programming relaxations for solving valued constraint satisfaction problems to optimality. The utility of linear programming relaxations in this context have previously been demonstrated using the lowest possible level of this hierarchy under the name of the basic linear programming relaxation (BLP). It has been shown that valued constraint languages containing only finite-valued weighted relations are tractable if, and only if, the integrality gap of the BLP is 1. In this paper, we demonstrate that almost all of the known tractable languages with arbitrary weighted relations have an integrality gap 1 for the Sherali-Adams relaxation with parameters (2,3). The result is closely connected to the notion of bounded relational width for the ordinary constraint satisfaction problem and its recent characterisation.