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IEEE Transactions on Computers

Spin-Variable Reduction Method for Handling Linear Equality Constraints in Ising Machines

Tatsuhiko Shirai, Nozomu Togawa

Published Jan 24, 2023• 11 citations
Publisher

Abstract

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.

DOI
10.1109/tc.2023.3239539

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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 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.
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 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.
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 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.
Ising machines can find optimum or quasi-optimum solutions of combinatorial optimization problems efficiently and effectively. The graph coloring problem, which is one of the difficult combinatorial optimization problems, is to assign a color to each vertex of a graph such that no two vertices connected by an edge have the same color. Although methods to map the graph coloring problem onto the Ising model or quadratic unconstrained binary optimization (QUBO) model are proposed, none of them considers minimizing the number of colors. In addition, there is no Ising-machine-based method considering additional constraints in order to apply to practical problems. In this paper, we propose a mapping method of the graph coloring problem including minimizing the number of colors and additional constraints to the QUBO model. As well as the constraint terms for the graph coloring problem, we firstly propose an objective function term that can minimize the number of colors so that the number of used spins cannot increase exponentially. Secondly, we propose two additional constraint terms: One is that specific vertices have to be colored with specified colors; The other is that specific colors cannot be used more than the number of times given in advance. We theoretically prove that, if the energy of the proposed QUBO mapping is minimized, all the constraints are satisfied and the objective function is minimized. The result of the experiment using an Ising machine showed that the proposed method reduces the number of used colors by up to 75.1% on average compared to the existing baseline method when additional constraints are not considered. Considering the additional constraints, the proposed method can effectively find feasible solutions satisfying all the constraints.