Parallelizable Search-Space Decomposition for Large-Scale Combinatorial Optimization Problems Using Ising Machines

TL;DR

This work proposes a novel search-space decomposition method that leverages the inherent structure of variables to systematically reduce the size of the master problem and suggests that search-space decomposition is a promising strategy for efficiently solving large-scale combinatorial optimization problems.

Abstract

Combinatorial optimization problems are crucial in industry. However, many COPs are NP-hard, causing the search space to grow exponentially with problem size and rendering large-scale instances computationally intractable. Conventional solvers typically treat problems as monolithic entities, leading to significant efficiency degradation as structural complexity increases. To address this issue, we propose a novel search-space decomposition method that leverages the inherent structure of variables to systematically reduce the size of the master problem. We formulate interaction costs between variables and individual variable costs as a constrained maximum cut problem and convert it into a quadratic unconstrained binary optimization formulation using penalty terms. An Ising-model solver is used to rapidly decompose the problem into independent small-scale subproblems, which are subsequently solved in parallel using mathematical optimization solvers. We validated this method on the capacitated vehicle routing problem. Results demonstrate three significant benefits: a substantial enhancement in feasible solution rates, accelerated convergence, achieving in 1 min the accuracy that the naive method required 30 min to reach, and a variable reduction of up to 95.32\%. These findings suggest that search-space decomposition is a promising strategy for efficiently solving large-scale combinatorial optimization problems.

Figures (10)

• Figure 1: Overview of the proposed method. Circles represent customers and the star represents the depot. (a) Master problem. (b) Partitioning of variables into red and blue sets. (c) Parallel solution of subproblems using a mathematical optimization solver. (d) Integration of subproblem solutions. Reprinted with permission from qce25_kawase, © 2025 IEEE.
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• Figure 6: Values of the dissimilarity measure $T_{i,j}$ as a function of $\theta_i$ and $\theta_j$.