The power of Sherali-Adams relaxations for general-valued CSPs

TL;DR

This paper provides an algebraic characterization of Sherali-Adams relaxations' effectiveness for solving valued CSPs, leading to new tractability results and simplified algorithms, along with a dichotomy theorem for certain classes of these problems.

Contribution

It offers a precise algebraic condition for the power of Sherali-Adams relaxations in valued CSPs, enabling simpler algorithms and new complexity classifications.

Findings

Several valued constraint languages are tractable via the third Sherali-Adams level.

A dichotomy theorem is established for languages expressing an injective unary function.

Simplified algorithms are derived for known languages compared to previous methods.

Abstract

We give a precise algebraic characterisation of the power of Sherali-Adams relaxations for solvability of valued constraint satisfaction problems to optimality. The condition is that of bounded width which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs. Our characterisation has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and many known valued constraint languages are tractable via the third level of the Sherali-Adams relaxation. For the known languages, this is a significantly simpler algorithm than the previously obtained ones. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov and Zivny [JACM'13], and also a dichotomy theorem for the exact solvability of Minimum-Solution problems. These are generalisations of Minimum-Ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and Uppman [ICALP'13].