The Houdayer Algorithm: Overview, Extensions, and Applications

TL;DR

This paper reviews the Houdayer algorithm, introduces a generalized version that explores many configurations efficiently, and demonstrates its effectiveness in sampling and optimization tasks related to Ising models and spin glasses.

Contribution

The paper presents a novel generalization of the Houdayer algorithm that exponentially increases configuration exploration while maintaining adaptability and low computational overhead.

Findings

Enhanced sampling efficiency in Ising models across various graphs

Successful application to binary optimization problems

Improved exploration of energy-preserving configurations

Abstract

The study of spin systems with disorder and frustration is known to be a computationally hard task. Standard heuristics developed for optimizing and sampling from general Ising Hamiltonians tend to produce correlated solutions due to their locality, resulting in a suboptimal exploration of the search space. To mitigate these effects, cluster Monte-Carlo methods are often employed as they provide ways to perform non-local transformations on the system. In this work, we investigate the Houdayer algorithm, a cluster Monte-Carlo method with small numerical overhead which improves the exploration of configurations by preserving the energy of the system. We propose a generalization capable of reaching exponentially many configurations at the same energy, while offering a high level of adaptability to ensure that no biased choice is made. We discuss its applicability in various contexts, including Markov chain Monte-Carlo sampling and as part of a genetic algorithm. The performance of our generalization in these settings is illustrated by sampling for the Ising model across different graph connectivities and by solving instances of well-known binary optimization problems. We expect our results to be of theoretical and practical relevance in the study of spin glasses but also more broadly in discrete optimization, where a multitude of problems follow the structure of Ising spin systems.