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.

Annealing machines such as quantum annealing machines and semiconductor-based annealing machines have been attracting attention as an efficient computing alternative for solving combinatorial optimization problems. They solve original combinatorial optimization problems by transforming them into a data structure called an Ising model. At that time, the bit-widths of the coefficients of the Ising model have to be kept within the range that an annealing machine can deal with. However, by reducing the Ising-model bit-widths, its minimum energy state, or ground state, may become different from that of the original one, and hence the targeted combinatorial optimization problem cannot be well solved. This paper proposes an effective method for reducing Ising model's bit-widths. The proposed method is composed of two processes: First, given an Ising model with large coefficient bit-widths, the shift method is applied to reduce its bit-widths roughly. Second, the spin-adding method is applied to further reduce its bit-widths to those that annealing machines can deal with. Without adding too many extra spins, we efficiently reduce the coefficient bit-widths of the original Ising model. Furthermore, the ground state before and after reducing the coefficient bit-widths is not much changed in most of the practical cases. Experimental evaluations demonstrate the effectiveness of the proposed method, compared to existing methods.

The binary quadratic knapsack problem (QKP) aims at optimizing a quadratic cost function within a single knapsack. Its applications and difficulty make it appealing for various industrial fields. In this paper we present an efficient strategy to solve the problem by modeling it as an Ising spin model using an Ising machine to search for its ground state which translates to the optimal solution of the problem. Secondly, in order to facilitate the search, we propose a novel technique to visualize the landscape of the search and demonstrate how difficult it is to solve QKP on an Ising machine. Finally, we propose two software solution improvement algorithms to efficiently solve QKP on an Ising machine.

In an amusement park, an attraction-visiting route considering the waiting time and traveling time improves visitors' satisfaction and experience. We focus on Ising machines to solve the problem, which are recently expected to solve combinatorial optimization problems at high speed by mapping the problems to Ising models or quadratic unconstrained binary optimization (QUBO) models. We propose a mapping of the visiting-route recommendation problem in amusement parks to a QUBO model for solving it using Ising machines. By using an actual Ising machine, we could obtain feasible solutions one order of magnitude faster with almost the same accuracy as the simulated annealing method for the visiting-route recommendation problem.

Ising machines have attracted attention as they are expected to solve combinatorial optimization problems at high speed with Ising models corresponding to those problems. An induced subgraph isomorphism problem is one of the decision problems, which determines whether a specific graph structure is included in a whole graph or not. The problem can be represented by equality constraints in the words of combinatorial optimization problem. By using the penalty functions corresponding to the equality constraints, we can utilize an Ising machine to the induced subgraph isomorphism problem. The induced subgraph isomorphism problem can be seen in many practical problems, for example, finding out a particular malicious circuit in a device or particular network structure of chemical bonds in a compound. However, due to the limitation of the number of spin variables in the current Ising machines, reducing the number of spin variables is a major concern. Here, we propose an efficient Ising model mapping method to solve the induced subgraph isomorphism problem by Ising machines. Our proposed method theoretically solves the induced subgraph isomorphism problem. Furthermore, the number of spin variables in the Ising model generated by our proposed method is theoretically smaller than that of the conventional method. Experimental results demonstrate that our proposed method can successfully solve the induced subgraph isomorphism problem by using the Ising-model based simulated annealing and a real Ising machine.

Recently, new optimization machines based on non-silicon physical systems, such as quantum annealing machines, have been developed, and their commercialization has been started. These machines solve the problems by searching the state of the Ising spins, which minimizes the Ising Hamiltonian. Such a property of minimization of the Ising Hamiltonian can be applied to various combinatorial optimization problems. In this paper, we introduce the coherent Ising machine (CIM), which can solve the problems in a milli-second order, and has higher performance than the quantum annealing machines especially on the problems with dense mutual connections in the corresponding Ising model. We explain how a target problem can be implemented on the CIM, based on the optimization scheme using the mutually connected neural networks. We apply the CIM to traveling salesman problems as an example benchmark, and show experimental results of the real machine of the CIM. We also apply the CIM to several combinatorial optimization problems in wireless communication systems, such as channel assignment problems. The CIM's ultra-fast optimization may enable a real-time optimization of various communication systems even in a dynamic communication environment.

Ising machines have attracted attention, which is expected to obtain better solutions of various combinatorial optimization problems at high speed by mapping the problems to natural phenomena. A slot-placement problem is one of the combinatorial optimization problems, regarded as a quadratic assignment problem, which relates to the optimal logic-block placement in a digital circuit as well as optimal delivery planning. Here, we propose a mapping to the Ising model for solving a slot-placement problem with additional constraints, called a constrained slot-placement problem, where several item pairs must be placed within a given distance. Since the behavior of Ising machines is stochastic and we map the problem to the Ising model which uses the penalty method, the obtained solution does not always satisfy the slot-placement constraint, which is different from the conventional methods such as the conventional simulated annealing. To resolve the problem, we propose an interpretation method in which a feasible solution is generated by post-processing procedures. We measured the execution time of an Ising machine and compared the execution time of the simulated annealing in which solutions with almost the same accuracy are obtained. As a result, we found that the Ising machine is faster than the simulated annealing that we implemented.