Large-scale quadratic unconstrained binary optimization (QUBO) formulations of constrained combinatorial optimization problems often exceed the input-size limit of present Ising machines or suffer from degraded solution quality as the number of binary variables increases. Large neighborhood search (LNS) mitigates this difficulty by sequentially optimizing restricted subproblems, but the structural factors that distinguish subproblems beyond the number of binary variables remain insufficiently characterized. In this study, we examine vehicle routing problems and compare a construction based on the vehicle routes of the current solution, denoted by LNS-K, with a construction based on QUBO variables and constraint relations, denoted by LNS-Q, while controlling the number of binary variables in the subproblems. Under the tested conditions, LNS-K obtained shorter total distances than LNS-Q in the matched-size comparisons, and the position variance, a measure of the spatial spread of the selected customers, decreased during the iterations in LNS-K. These observations suggest that subproblem design for sequential optimization with Ising machines should consider not only subproblem size but also semantic and geometric structures inherited from the current solution.
Large-scale quadratic unconstrained binary optimization (QUBO) formulations of constrained combinatorial optimization problems often exceed the input-size limit of present Ising machines or suffer from degraded solution quality as the number of binary variables increases. Large neighborhood search (LNS) mitigates this difficulty by sequentially optimizing restricted subproblems, but the structural factors that distinguish subproblems beyond the number of binary variables remain insufficiently characterized. In this study, we examine vehicle routing problems and compare a construction based on the vehicle routes of the current solution, denoted by LNS-K, with a construction based on QUBO variables and constraint relations, denoted by LNS-Q, while controlling the number of binary variables in the subproblems. Under the tested conditions, LNS-K obtained shorter total distances than LNS-Q in the matched-size comparisons, and the position variance, a measure of the spatial spread of the selected customers, decreased during the iterations in LNS-K. These observations suggest that subproblem design for sequential optimization with Ising machines should consider not only subproblem size but also semantic and geometric structures inherited from the current solution.
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.
The quadratic unconstrained binary optimization (QUBO) problem is categorized as an NP-hard combinatorial optimization problem. The variable neighborhood search (VNS) algorithm is one of the leading algorithms used to solve QUBO problems. As neighborhood structure change is the central concept in the VNS algorithm, the design of the neighborhood structure is crucial. This paper presents a modified VNS algorithm called “B-VNS”, which can be used to solve QUBO problems. A binomial trial was used to construct the neighborhood structure, and this was used with the aim of reducing computation time. The B-VNS and VNS algorithms were tested on standard QUBO problems from Glover and Beasley, on standard max-cut problems from Helmberg–Rendl, and on those proposed by Burer, Monteiro, and Zhang. Finally, Mann–Whitney tests were conducted using α=0.05, to statistically compare the performance of the two algorithms. It was shown that the B-VNS and VNS algorithms are able to provide good solutions, but the B-VNS algorithm runs substantially faster. Furthermore, the B-VNS algorithm performed the best in all of the max-cut problems, regardless of problem size, and it performed the best in QUBO problems, with sizes less than 500. The results suggest that the use of binomial distribution, to construct the neighborhood structure, has the potential for further development.
Ising machines are next-generation computers expected to efficiently sample near-optimal solutions of combinatorial optimization problems. Combinatorial optimization problems are modeled as quadratic unconstrained binary optimization (QUBO) problems to apply an Ising machine. However, current state-of-the-art Ising machines still often fail to output nearoptimal solutions due to the complicated energy landscape of QUBO problems. Furthermore, the physical implementation of Ising machines severely restricts the size of QUBO problems to be input as a result of limited hardware graph structures. In this study, we take a new approach to these challenges by injecting auxiliary penalties preserving the optimum, which reduces quadratic terms in QUBO objective functions. The process simultaneously simplifies the energy landscape of QUBO problems, allowing the search for near-optimal solutions, and makes QUBO problems sparser, facilitating encoding into Ising machines with restriction on the hardware graph structure. We propose linearization of QUBO problems using variable posets as an outcome of the approach. By applying the proposed method to synthetic QUBO instances and to multi-dimensional knapsack problems, we empirically validate the effects on enhancing minorembedding of QUBO problems and the performance of Ising machines.
Ising machines are specialized solvers for combinatorial optimization problems (COPs), which are typically formulated as quadratic unconstrained binary optimization (QUBO) models. To enhance their performance, various methods have been proposed to construct subQUBOs by reducing the number of variables in the original QUBO. This study proposes a general subQUBO construction method applicable to a wide range of problems. The variables are clustered based on their interaction structure using theK-means clustering. A subQUBO is constructed from a randomly selected cluster and a pre-obtained tentative solution and then solved. This process of cluster selection and subQUBOs is performed iteratively. We evaluated the performance of the method on the quadratic assignment problem and compared it with two baselines: one that randomly reduces variables and another that directly solves the original QUBO without any variable reduction. Experimental evaluations demonstrated that the proposed method achieves high-accuracy solutions within the same computation time, particularly for large problem sizes.
Large-scale combinatorial optimization is a challenge for near-term quantum computing because dense Quadratic Unconstrained Binary Optimization (QUBO) formulations yield interaction graphs that exceed the limits of NISQ processors. This work introduces Hybrid Quantum Neighborhood Selection (HQNS), a hybrid framework mitigating this via stochastic frontier decomposition. Instead of encoding all N variables into a monolithic circuit, HQNS selects a compact frontier of F << N active variables per stage, freezing the rest into reduced QUBO coefficients. A multi-stage crawling procedure rotates these frontiers, letting local quantum subproblems refine a global solution. We evaluate HQNS on the Maximum Diversity Subset Selection Problem (MDSSP) across six scales, N up to 1000. Circuit burden is reduced from the dense QAOA requirement of O(N^2) two-qubit terms per layer to O(F^2) per stage, with total complexity governed by the number of stages and classical overhead. Benchmarks show that HQNS achieves competitive solution quality relative to parallel simulated annealing (SA) while maintaining bounded circuit width and stable QPU time. In the N=1000 benchmark over ten executions, HQNS preserves 99.9908% of the mean diversity score of an 11-restart parallel SA baseline, while reducing wall-clock time by 65.03%, peak CPU usage by 55.97%, and peak memory by 35.21%. Ablation shows performance depends on frontier size, warm-starts, CVaR filtering, and stochastic rotation. These results demonstrate that structured frontier decomposition makes variational optimization executable for dense QUBO instances unsuitable for direct QAOA on present hardware.
We present a heuristic algorithm designed to solve Quadratic Unconstrained Binary Optimization (QUBO) problems efficiently. The algorithm, referred to as IC-D2S, leverages a hybrid approach using Ising and classical machines to address very large problem sizes. Considering the practical limitation on the size of the Ising machine (IM), our algorithm partitions the QUBO problem into a collection of QUBO subproblems (called subQUBOs) and utilizes the IM to solve each subQUBO. Our proposed heuristic algorithm uses a set of control parameters to generate the subQUBOs and explore the search space. Also, it utilizes an annealer based on cosine waveform and applies a mutation operator at each step of the search to diversify the solution space and facilitate the process of finding the global minimum of the problem. We have evaluated the effectiveness of our IC-D2S algorithm on three large-sized problem sets and compared its efficiency in finding the (near-)optimal solution with three QUBO solvers. One of the solvers is a software-based algorithm (D2TS), while the other one (D-Wave) employs a similar approach to ours, utilizing both classical and Ising machines. The results demonstrate that for large-sized problems (≥ 5000) the proposed algorithm identifies superior solutions. Additionally, for smaller-sized problems (= 2500), IC-D2S efficiently finds the optimal solution in a significantly faster manner.