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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Ising-Machine-Based Solver for Constrained Graph Coloring Problems

Soma Kawakami, Yosuke Mukasa, Siya Bao, Dema BA, Junya Arai, Satoshi Yagi, Junji Teramoto, Nozomu Togawa

Published Sep 11, 2023• 3 citations
Publisher

Abstract

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.

DOI
10.1587/transfun.2023kep0003

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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.
Optimum or quasi-optimum solutions of combinatorial optimization problems can be efficiently found by using Ising machines. The graph coloring problem (GCP) is known as a difficult combinatorial problem. Given a graph, each vertex should be assigned a color and any two vertices connected by an edge must not be colored the same. Although methods to map the GCP onto the Ising model are proposed, none of them considers minimizing the number of colors. In this paper, we propose a mapping method of the GCP including minimizing the number of colors and additional constraints to the QUBO (Quadratic Unconstrained Binary Optimization) model. As well as the constraint terms for the GCP, 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 more additional terms so that our proposed method can be applicable to practical consumer applications. The experimental evaluations on an Ising machine showed that the proposed method reduces the number of used colors by up to 75.1% on average compared with the existing baseline method. Considering the additional constraints, the proposed method can effectively find feasible solutions satisfying all the constraints.
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 are efficient solvers for combinatorial optimization problems (COPs). After mapping COPs to Quadratic Unconstrained Binary Optimization (QUBO) problems, Ising machines search for the lowest energy solution of the QUBO corresponding to the optimal solution of the COP. Herein we propose an efficient method to formulate a lecturer–student assignment problem as a QUBO problem. In this problem, the sets of subjects, lecturers, students, lecturers’ teachable subjects, and students’ required subjects are given. To minimize lecture costs, the number of lecturers in charge of students must be minimized. The number of combinations drastically increases as the number of lecturers or students increases. Using these results, we theoretically prove the optimality, i.e., the formulated QUBO takes the lowest energy if and only if the students are optimally assigned to the lecturers. Additionally, we demonstrate that an Ising machine outperforms simulated annealing in terms of solution quality and execution time.
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 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 emerging hardware architectures that efficiently solve NP-Hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows ~10000x performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5-4x over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.