  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.
