Pliability and Approximating Max-CSPs

TL;DR

This paper introduces the concept of treewidth-pliability, providing a unified polynomial-time approximation scheme for a broad class of Max-2-CSPs and related problems, extending existing techniques to new graph classes.

Contribution

It defines treewidth-pliability as a sufficient condition for approximation, unifying Baker's layering and Szemerédi's regularity lemma approaches, and explores its connections to graph theory concepts.

Findings

Provides a polynomial-time approximation scheme for Max-2-CSPs under treewidth-pliability.

Extends applicability of approximation techniques to new classes of graphs.

Shows the condition cannot be used for exact solutions in general.

Abstract

We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker's layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemer\'edi's regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. On the other hand, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.