PTAS for Sparse General-Valued CSPs

TL;DR

This paper develops polynomial-time approximation schemes (PTAS) for a broad class of constraint satisfaction problems on sparse graphs, extending existing methods to general-valued CSPs with crisp constraints under relaxed conditions.

Contribution

It extends PTAS results to general-valued CSPs with crisp constraints, relaxing previous conditions to diagonalisability, and broadening applicability to more graph classes.

Findings

PTAS for minimisation general-valued CSPs on Baker graph classes

Extension of Sherali-Adams LP relaxation to general-valued CSPs

Relaxation of conditions from fractionally-treewidth-fragile to diagonalisability

Abstract

We study polynomial-time approximation schemes (PTASes) for constraint satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex Cover on sparse graph classes. Baker's approach gives a PTAS on planar graphs, excluded-minor classes, and beyond. For Max-CSPs, and even more generally, maximisation finite-valued CSPs (where constraints are arbitrary non-negative functions), Romero, Wrochna, and \v{Z}ivn\'y [SODA'21] showed that the Sherali-Adams LP relaxation gives a simple PTAS for all fractionally-treewidth-fragile classes, which is the most general "sparsity" condition for which a PTAS is known. We extend these results to general-valued CSPs, which include "crisp" (or "strict") constraints that have to be satisfied by every feasible assignment. The only condition on the crisp constraints is that their domain contains an element which is at least as feasible as all the others (but possibly less valuable). For minimisation general-valued CSPs with crisp constraints, we present a PTAS for all Baker graph classes -- a definition by Dvo\v{r}\'ak [SODA'20] which encompasses all classes where Baker's technique is known to work, except possibly for fractionally-treewidth-fragile classes. While this is standard for problems satisfying a certain monotonicity condition on crisp constraints, we show this can be relaxed to diagonalisability -- a property of relational structures connected to logics, statistical physics, and random CSPs.