Single-Temperature Ranked Greedy Cluster Exchange for Max-Cut on Two-Dimensional Tori

Abstract

Ground-state search in two-dimensional Ising spin glasses on large toroidal lattices remains challenging for local-update heuristics because rugged energy barriers impede relaxation across metastable states. Equivalent to zero-field toroidal Max-Cut problems, these spin-glass instances also serve as standard benchmarks for optimization heuristics. Many Monte Carlo approaches combine single-spin flips or cluster moves with temperature ladders or annealing schedules, yet even with tuned parameters, rarely achieve rapid, reliable ground-state convergence at scale. Here, we address the need for faster ground-state search through cooperation between replicas. We introduce ranked greedy cluster exchange (RGCX) and combine it with fixed-temperature Metropolis sweeps to form single-temperature RGCX. Replicas are ranked by cut to determine the pair-processing order. Each pair deterministically exchanges every disagreement cluster that raises the better replica's cut, retaining the complement and conserving the pair's summed cut. On the 20,000-spin Gset instance G81, the estimated runtime to reach the certified optimum with 99% success probability (TTS99) was 12.1 s on an 8-core desktop computer. With the same population size, exchange interval and temperature, the method reached the best known cuts on all 17 toroidal Gset graphs, including 9 certified optima. These findings advance ground-state search on large two-dimensional tori by establishing single-temperature RGCX as an efficient population-based method that combines ranked greedy cluster exchange with fixed-temperature Metropolis dynamics.