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.

Ising machines can find optimum or quasi-optimum solutions of combinatorial optimization problems efficiently and effectively. It is known that, when a good initial solution is given to an Ising machine, we can finally obtain a solution closer to the optimal solution. However, several Ising machines cannot directly accept an initial solution due to its computational nature. In this paper, we propose a method to give quasi-initial solutions into Ising machines that cannot directly accept them. The proposed method gives the positive or negative external magnetic field coefficients (magnetic field controlling term) based on the initial solutions and obtains a solution by using an Ising machine. Then, the magnetic field controlling term is re-calculated every time an Ising machine repeats the annealing process, and hence the solution is repeatedly improved on the basis of the previously obtained solution. The proposed method is applied to the capacitated vehicle routing problem with an additional constraint (constrained CVRP) and the max-cut problem. Experimental results show that the total path distance is reduced by 5.78% on average compared to the initial solution in the constrained CVRP and the sum of cut-edge weight is increased by 1.25% on average in the max-cut problem.