The Houdayer Algorithm: Overview, Extensions, and Applications

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.