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Efficient algorithms for the Potts model on small-set expanders

Charles Carlson, Ewan Davies, Alexandra Kolla

TL;DR

This work tackles the problem of approximating the ferromagnetic Potts partition function $Z_G(\beta)$ in the low-temperature regime on graphs with small-set expansion. It develops a framework combining partitioning of the graph into expanders, a decomposition into pseudo-ground states, and a cluster-expansion-based polymer model to efficiently approximate contributions from near-ground configurations. The main contributions include an explicit FPTAS under a small-set expansion condition for $\Delta$-regular graphs (and a corresponding irregular-graph variant with a spectral expansion condition), plus an algorithmic blueprint that connects small-set expansion, spectral gaps, and Unique Games techniques to approximate counting problems beyond standard expanders. The results advance subexponential-time approaches for #BIS and illuminate deep links between spectral graph properties and approximate counting methods, while suggesting several natural directions for extending these techniques to broader graph classes and related spin systems.

Abstract

An emerging trend in approximate counting is to show that certain `low-temperature' problems are easy on typical instances, despite worst-case hardness results. For the class of regular graphs one usually shows that expansion can be exploited algorithmically, and since random regular graphs are good expanders with high probability the problem is typically tractable. Inspired by approaches used in subexponential-time algorithms for Unique Games, we develop an approximation algorithm for the partition function of the ferromagnetic Potts model on graphs with a small-set expansion condition. In such graphs it may not suffice to explore the state space of the model close to ground states, and a novel feature of our method is to efficiently find a larger set of `pseudo-ground states' such that it is enough to explore the model around each pseudo-ground state.

Efficient algorithms for the Potts model on small-set expanders

TL;DR

This work tackles the problem of approximating the ferromagnetic Potts partition function in the low-temperature regime on graphs with small-set expansion. It develops a framework combining partitioning of the graph into expanders, a decomposition into pseudo-ground states, and a cluster-expansion-based polymer model to efficiently approximate contributions from near-ground configurations. The main contributions include an explicit FPTAS under a small-set expansion condition for -regular graphs (and a corresponding irregular-graph variant with a spectral expansion condition), plus an algorithmic blueprint that connects small-set expansion, spectral gaps, and Unique Games techniques to approximate counting problems beyond standard expanders. The results advance subexponential-time approaches for #BIS and illuminate deep links between spectral graph properties and approximate counting methods, while suggesting several natural directions for extending these techniques to broader graph classes and related spin systems.

Abstract

An emerging trend in approximate counting is to show that certain `low-temperature' problems are easy on typical instances, despite worst-case hardness results. For the class of regular graphs one usually shows that expansion can be exploited algorithmically, and since random regular graphs are good expanders with high probability the problem is typically tractable. Inspired by approaches used in subexponential-time algorithms for Unique Games, we develop an approximation algorithm for the partition function of the ferromagnetic Potts model on graphs with a small-set expansion condition. In such graphs it may not suffice to explore the state space of the model close to ground states, and a novel feature of our method is to efficiently find a larger set of `pseudo-ground states' such that it is enough to explore the model around each pseudo-ground state.

Paper Structure

This paper contains 17 sections, 15 theorems, 109 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Connections to Unique Games
  3. Preliminaries
  4. Our results
  5. Technical overview
  6. Related work
  7. Further questions
  8. Organisation
  9. Approximating Potts partition functions
  10. Dealing with the bad parts
  11. Dealing with the good parts
  12. Approximation by states close to a ground state
  13. Approximation by a polymer model
  14. Convergence of the cluster expansion
  15. Finishing the argument for good parts
  16. ...and 2 more sections

Key Result

Theorem 1

Let $G=(V,E)$ be a $\Delta$-regular graph on $n$ vertices. There is an absolute constant $C$ such that the following holds. Suppose that for some integer $k\ge 2$, we have the small-set expansion condition that for all sets $S\subset V(G)$ of size at most $n/k$. Then for the $q$-color ferromagnetic Potts model with there is a deterministic algorithm that produces a relative $\varepsilon$-approxi

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof
  • Theorem 8: Jenssen, Keevash, and Perkins JKP19
  • Theorem 9
  • Theorem 10: Lee, Oveis Gharan and Trevisan LOT12
  • ...and 22 more