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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.

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E107-A No.1 pp.38-51

- Publication Date
- 2024/01/01

- Publicized
- 2023/09/12

- Online ISSN
- 1745-1337

- DOI
- 10.1587/transfun.2023KEP0003

- Type of Manuscript
- Special Section PAPER (Special Section on Circuits and Systems)

- Category

Soma KAWAKAMI

Waseda University

Yosuke MUKASA

Waseda University

Siya BAO

Waseda University

Dema BA

Nippon Telegraph and Telephone Corporation

Junya ARAI

Nippon Telegraph and Telephone Corporation

Satoshi YAGI

Nippon Telegraph and Telephone Corporation

Junji TERAMOTO

Nippon Telegraph and Telephone Corporation

Nozomu TOGAWA

Waseda University

The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.

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Soma KAWAKAMI, Yosuke MUKASA, Siya BAO, Dema BA, Junya ARAI, Satoshi YAGI, Junji TERAMOTO, Nozomu TOGAWA, "Ising-Machine-Based Solver for Constrained Graph Coloring Problems" in IEICE TRANSACTIONS on Fundamentals,
vol. E107-A, no. 1, pp. 38-51, January 2024, doi: 10.1587/transfun.2023KEP0003.

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.

URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2023KEP0003/_p

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@ARTICLE{e107-a_1_38,

author={Soma KAWAKAMI, Yosuke MUKASA, Siya BAO, Dema BA, Junya ARAI, Satoshi YAGI, Junji TERAMOTO, Nozomu TOGAWA, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={Ising-Machine-Based Solver for Constrained Graph Coloring Problems},

year={2024},

volume={E107-A},

number={1},

pages={38-51},

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.},

keywords={},

doi={10.1587/transfun.2023KEP0003},

ISSN={1745-1337},

month={January},}

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TY - JOUR

TI - Ising-Machine-Based Solver for Constrained Graph Coloring Problems

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 38

EP - 51

AU - Soma KAWAKAMI

AU - Yosuke MUKASA

AU - Siya BAO

AU - Dema BA

AU - Junya ARAI

AU - Satoshi YAGI

AU - Junji TERAMOTO

AU - Nozomu TOGAWA

PY - 2024

DO - 10.1587/transfun.2023KEP0003

JO - IEICE TRANSACTIONS on Fundamentals

SN - 1745-1337

VL - E107-A

IS - 1

JA - IEICE TRANSACTIONS on Fundamentals

Y1 - January 2024

AB - 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.

ER -