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International Conference on Consumer Electronics

Constraint-Parameterized Spin-Variable Reduction Method for QAP

Raio Aoki, Takeru Ota, Tatsuhiko Shirai, Nozomu Togawa

Published Feb 3, 2026• 0 citations
Publisher

Abstract

Developing novel quadratic unconstrained binary optimization (QUBO) formulations is important to enhance the performance of the Ising machines. The spin-variable reduction (SVR) method formulates the QUBO with fewer binary variables than the penalty method. However, the SVR method can complicate the energy landscape. To address this issue, we propose a constraint-parameterized SVR method for the combinatorial optimization problems with a 2-way 1-hot constraint such as the quadratic assignment problem (QAP). Our proposed method introduces two key elements: (1) preliminary analysis using simulated annealing (SA) to statistically identify the variable that has a high probability of being the value of 1, and (2) parameterization of the SVR constraint term using this statistical information. Experiments on QAP benchmarks show that our method can find higher-precision solutions more efficiently than conventional SVR or the penalty method in certain cases.

DOI
10.1109/icce67443.2026.11449668

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Developing novel quadratic unconstrained binary optimization (QUBO) formulations is important to enhance the performance of the Ising machines. The spin-variable reduction (SVR) method formulates the QUBO with fewer binary variables than the penalty method. However, the SVR method can complicate the energy landscape. To address this issue, we propose a constraint-parameterized SVR method for the combinatorial optimization problems with a 2-way 1-hot constraint such as the quadratic assignment problem (QAP). Our proposed method introduces two key elements: (1) preliminary analysis using simulated annealing (SA) to statistically identify the variable that has a high probability of being the value of 1, and (2) parameterization of the SVR constraint term using this statistical information. Experiments on QAP benchmarks show that our method can find higher-precision solutions more efficiently than conventional SVR or the penalty method in certain cases.
Ising machines, including quantum annealers, have attracted much attention as an efficient solver for COPs (Combinatorial optimization problems). The SVR (Spin-Variable Reduction) method maps a COP with linear equality constraints to QUBO (Quadratic Unconstrained Binary Optimization) model, which is a common input for using the Ising machines. In this paper, we conduct a tuning of a constraint coefficient for this method using Quadratic Assignment Problems as a case study. We find that the optimal value of the constraint coefficient can be either larger or smaller than that used in the conventional penalty method, depending on the problem instance. We discuss the underlying mechanism responsible for this difference.
We propose a spin-variable reduction method for Ising machines to handle linear equality constraints in a combinatorial optimization problem. Ising machines including quantum-annealing machines can effectively solve combinatorial optimization problems. They are designed to find the lowest-energy solution of a quadratic unconstrained binary optimization (QUBO), which is mapped from the combinatorial optimization problem. The proposed method reduces the number of binary variables to formulate the QUBO compared to the conventional penalty method. We demonstrate a sufficient condition to obtain the optimum of the combinatorial optimization problem in the spin-variable reduction method and its general applicability. We apply it to typical combinatorial optimization problems, such as the graph κ-partitioning problem and the quadratic assignment problem. Experiments using simulated-annealing and quantum-annealing based Ising machines demonstrate that the spin-variable reduction method outperforms the penalty method. The proposed method extends the application of Ising machines to larger-size combinatorial optimization problems with linear equality constraints.
Ising machines including quantum annealers have attracted attention as efficient solvers for combinatorial optimization problems (COPs). To use Ising machines, the objective function and constraints of a COP must be expressed in a QUBO (quadratic unconstrained binary optimization) form. However, their application range is practically limited since some COPs are difficult to formulate in a QUBO form, or require a large number of binary variables. Quadratic three-dimensional assignment problem (Q3AP) is a challenging problem for Ising machines to solve since the number of required binary variables rapidly increases with the problem size N. In this paper, to overcome the difficulty, we propose a method to reduce the number of necessary variables fromN3N^{3}N3to2N22N^{2}2N2by using a factorization machine with quantum annealing (FMQA) to estimate QUBO through machine learning. Experimental results with a real Ising machine demonstrate that the proposed method can obtain the best-known solution in a short time.
Many combinatorial optimization problems can be formulated as a problem to determine the order of sequence or to find a corresponding mapping of the objects. We call such problems permutation-based optimization problems. Many such problems can be formulated as a quadratic unconstrained binary optimization (QUBO) or Ising model by introducing a penalty coefficient to the permutation constraint terms. While classical and quantum annealing approaches have been proposed to solve QUBOs to date, they face issues with optimality and feasibility. Here we treat a given QUBO solver as a black box and propose techniques to enhance its performance. First, to ensure an effective search for good quality solutions, a smooth energy landscape is needed; we propose a data scaling approach that reduces the amplitudes of the input without compromising optimality. Second, we need to tune the penalty coefficient. In this paper, we illustrate that for certain problems, it suffices to tune the parameter by performing random sampling on the penalty coefficients to achieve good performance. Finally, to handle possible infeasibility of the solution, we introduce a polynomial-time projection algorithm. We apply these techniques along with a divide-and-conquer strategy to solve some large-scale permutation-based problems and present results for TSP and QAP.
In combinatorial optimization problems with one-hot constraints, the number of binary variables that have the value of one in the feasible solution becomes constant and small, so it is important to always consider them to escalate the solution searching process and obtain a more converged solution in annealing. In this paper, we propose a method to preferably select binary variables with the value of one in the subQUBO annealing process using Ising machines such as quantum annealers. Evaluation experiments show that the proposed method obtains a solution with lower energy than a conventional subQUBO method and a full QUBO method.
Ising machines have attracted much attention in recent years because of their potential in solving combinatorial optimization problems that are difficult to be solved using conventional computers. Ising machines solve the optimization problem that can be transformed into a quadratic unconstrained binary optimization (QUBO) problem. In the case of a problem with constraints, named a constraint problem, an objective function and constraint function with penalty coefficients are combined into a single Hamiltonian of a QUBO problem. When solving a constraint problem, the accuracy of the solution decreases compared to a QUBO problem with an objective function only. This is because penalty coefficients are set to excessively large values to prevent constraint violations. As a result, it becomes difficult to minimize the objective function. To solve this problem, this paper proposes a method that partitions constraint functions and reduces the value of the penalty coefficients. The performance evaluation using the traveling salesperson problem (TSP) with one-hot constraints demonstrates that the proposed method can improve the solution accuracy compared to the conventional method.