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The antiferromagnetic Ising model beyond line graphs

Mark Jerrum

Abstract

Both the antiferromagnetic Ising model and the hard-core model could be said to be tractable on line graphs of bounded degree. For example, Glauber dynamics is rapidly mixing in both cases. In the case of the hard-core model, we know that tractability extends further, to claw-free graphs and somewhat beyond. In contrast, it is shown here that the corresponding extensions are not possible in the case of the antiferromagnetic Ising model.

The antiferromagnetic Ising model beyond line graphs

Abstract

Both the antiferromagnetic Ising model and the hard-core model could be said to be tractable on line graphs of bounded degree. For example, Glauber dynamics is rapidly mixing in both cases. In the case of the hard-core model, we know that tractability extends further, to claw-free graphs and somewhat beyond. In contrast, it is shown here that the corresponding extensions are not possible in the case of the antiferromagnetic Ising model.
Paper Structure (5 sections, 5 theorems, 24 equations, 1 figure, 1 algorithm)

This paper contains 5 sections, 5 theorems, 24 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Motivation
  2. Preliminaries
  3. Complexity-theoretic analysis
  4. Exponential-time mixing
  5. Interactions of varying strength

Key Result

Lemma 1

It is $\mathrm{NP}$-hard to approximate $\textsc{Cubic\-MaxCut}$ within ratio $0.997$. That is to say, it is $\mathrm{NP}$-hard to compute a number $c$ such that $0.997C\leq c\leq C$, where $C=\textsc{Cubic\-MaxCut}(G)$.

Figures (1)

  • Figure 1: An edge $e=\{u,v\}$ in $G$ and its translation in the graph $G^*$. The dashed 'cartouches' indicate cliques.

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5