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Pliability and Approximating Max-CSPs

Miguel Romero, Marcin Wrochna, Stanislav Živný

TL;DR

The paper introduces treewidth-pliability as a unifying condition under which Max-Hom and Max-CSPs admit PTASes across a wide spectrum of underlying graph classes, spanning sparse to dense regimes. By connecting pliability to the Sherali–Adams LP relaxations, it shows that any tw-pliable class yields a PTAS for Max-Hom, and it situates pliability within the broader landscape of fractional fragility, hyperfiniteness, and density-based regularity methods. The authors establish broad equivalences among sparsity notions (tw, td, Hadwiger, cc) and derive both positive results (dense and hyperfinite classes) and hardness boundaries (tournaments under Gap-ETH). They also discuss the limits of the framework, open questions, and potential extensions to EPTAS and property-testing-inspired methods, underscoring the framework’s theoretical breadth and practical implications for approximation in CSPs and homomorphism problems.

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 Szemerédi'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.

Pliability and Approximating Max-CSPs

TL;DR

The paper introduces treewidth-pliability as a unifying condition under which Max-Hom and Max-CSPs admit PTASes across a wide spectrum of underlying graph classes, spanning sparse to dense regimes. By connecting pliability to the Sherali–Adams LP relaxations, it shows that any tw-pliable class yields a PTAS for Max-Hom, and it situates pliability within the broader landscape of fractional fragility, hyperfiniteness, and density-based regularity methods. The authors establish broad equivalences among sparsity notions (tw, td, Hadwiger, cc) and derive both positive results (dense and hyperfinite classes) and hardness boundaries (tournaments under Gap-ETH). They also discuss the limits of the framework, open questions, and potential extensions to EPTAS and property-testing-inspired methods, underscoring the framework’s theoretical breadth and practical implications for approximation in CSPs and homomorphism problems.

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 Szemerédi'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.

Paper Structure

This paper contains 31 sections, 44 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sparse structures: Baker's technique and fragility
  3. Dense structures: the regularity lemma
  4. Robustness of pliability
  5. Related work
  6. Overview
  7. Preliminaries
  8. Structures
  9. Overcasts
  10. Pliability and graph parameters
  11. Opt-distance zero and edit-distance
  12. Pliable examples
  13. Non-pliable examples
  14. Pliable structures admit a PTAS: proof of Theorem \ref{['thm:main1']}
  15. Fractional fragility
  16. ...and 16 more sections

Key Result

Theorem 1.1

If $\mathcal{A}$ is a $\mathop{\mathrm{tw}}\nolimits$-pliable class of structures of bounded arity, then $\hbox{Max-Hom}\xspace(\mathcal{A},-)$ admits a PTAS.

Figures (1)

  • Figure 1: The Sherali-Adams relaxation of level $k$ of $\hbox{Max-Hom}\xspace$ instance $(\mathbb{A},\mathbb{B})$.

Theorems & Definitions (103)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 93 more