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Semidefinite programming and linear equations vs. homomorphism problems

Lorenzo Ciardo, Stanislav Živný

TL;DR

The paper develops a hybrid relaxation, SDA, that integrates semidefinite programming with a linear Diophantine solver to tackle graph-homomorphism-type problems. By leveraging the spectral theory of association schemes, it analyzes how symmetry shapes the relaxation's behavior and derives an unconditional lower bound showing SDA cannot solve the Approximate Graph Homomorphism problem, implying hardness for approximate graph coloring under standard assumptions. It introduces a matrix/orbital framework that translates SDP feasibility into linear constraints via character tables of association schemes, enabling explicit lower-bound constructions using Johnson and cycle schemes. The results situate SDA as strictly stronger than SDP alone but incomparable with the BA hierarchy, and they extend non-solvability results to the broader Approximate CSP/PCSP setting, with conditional hardness consequences via the Unique Games Conjecture. Overall, the work clarifies limits of SDP+linear-equation relaxations and provides a symmetry-aware methodology for proving lower bounds in highly structured instances.

Abstract

We introduce a relaxation for homomorphism problems that combines semidefinite programming with linear Diophantine equations, and propose a framework for the analysis of its power based on the spectral theory of association schemes. We use this framework to establish an unconditional lower bound against the semidefinite programming + linear equations model, by showing that the relaxation does not solve the approximate graph homomorphism problem and thus, in particular, the approximate graph colouring problem.

Semidefinite programming and linear equations vs. homomorphism problems

TL;DR

The paper develops a hybrid relaxation, SDA, that integrates semidefinite programming with a linear Diophantine solver to tackle graph-homomorphism-type problems. By leveraging the spectral theory of association schemes, it analyzes how symmetry shapes the relaxation's behavior and derives an unconditional lower bound showing SDA cannot solve the Approximate Graph Homomorphism problem, implying hardness for approximate graph coloring under standard assumptions. It introduces a matrix/orbital framework that translates SDP feasibility into linear constraints via character tables of association schemes, enabling explicit lower-bound constructions using Johnson and cycle schemes. The results situate SDA as strictly stronger than SDP alone but incomparable with the BA hierarchy, and they extend non-solvability results to the broader Approximate CSP/PCSP setting, with conditional hardness consequences via the Unique Games Conjecture. Overall, the work clarifies limits of SDP+linear-equation relaxations and provides a symmetry-aware methodology for proving lower bounds in highly structured instances.

Abstract

We introduce a relaxation for homomorphism problems that combines semidefinite programming with linear Diophantine equations, and propose a framework for the analysis of its power based on the spectral theory of association schemes. We use this framework to establish an unconditional lower bound against the semidefinite programming + linear equations model, by showing that the relaxation does not solve the approximate graph homomorphism problem and thus, in particular, the approximate graph colouring problem.
Paper Structure (14 sections, 34 theorems, 97 equations, 1 figure)

This paper contains 14 sections, 34 theorems, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work on association schemes
  3. Structure of the paper
  4. Notation
  5. The algorithms
  6. Automorphisms and orbitals
  7. Association schemes
  8. The Johnson scheme
  9. The cycle scheme
  10. A lower bound against SDA
  11. Incomparability to the BA hierarchy
  12. A tale of two polytopes
  13. SDP${}^{\epsilon}$ vs. BA${}^k$
  14. Appendix

Key Result

Theorem 1

Let $\mathbf{A},\mathbf{B}$ be non-bipartite loopless undirected graphs such that $\mathbf{A}\to\mathbf{B}$. Then $\mathop{\mathrm{SDA}}\nolimits$ does not solve $\mathop{\mathrm{PCSP}}\nolimits(\mathbf{A},\mathbf{B})$.

Figures (1)

  • Figure 1: From the left, the generalised Johnson graphs $\mathbf{J}_{7,3,1}$, $\mathbf{J}_{7,3,2}$, and $\mathbf{J}_{7,3,3}$.

Theorems & Definitions (65)

  • Conjecture : GJ76
  • Conjecture : BG21
  • Theorem 1
  • Definition 2
  • Proposition 3
  • Proposition 4
  • Lemma 5
  • Proposition 6
  • Proposition 7
  • Definition 8
  • ...and 55 more