A Multiple Coefficients Trial Method to Solve Combinatorial Optimization Problems for Simulatedannealingbased Ising Machines When solving a combinatorial optimization problem with Ising machines, we have to formulate it onto an energy function of Ising model. Here, how to determine the penalty coefficients in the energy function is a great concern if it includes constraint terms. In this paper, we focus on a traveling salesman problem (TSP, in short), one of the combinatorial optimization problems with equality constraints. Firstly, we investigate the relationship between the penalty coefficient and the accuracy of solutions in a TSP. Based on it, we propose a method to obtain a TSP quasioptimum solution, which is called multiple coefficients trial method. In our proposed method, we use an Ising machine to solve a TSP by changing a penalty coefficient every trial, the TSP solutions can converge very fast in total. Compared to naive methods using simulatedannealingbased Ising machines, we confirmed that our proposed method can reduce the total number of annealing iterations to 110 to 11000 to obtain a quasioptimum solution in 32city TSPs.

Mapping Constrained SlotPlacement Problems to Ising Models and its Evaluations by an Ising Machine 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 slotplacement problem is one of the combinatorial optimization problems, regarded as a quadratic assignment problem, which relates to the optimal logicblock placement in a digital circuit as well as optimal delivery planning. Here, we propose a mapping to the Ising model for solving a slotplacement problem with additional constraints, called a constrained slotplacement problem, where several item pairs must be placed within a given distance. Since the behavior of Ising machines is stochastic, the obtained solution does not always satisfy the slotplacement 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 postprocessing procedures. Using an Ising machine computer, feasible solutions could be obtained up to 50 times faster than the conventional simulated annealing without degrading accuracy for constrained slotplacement problems with 6 x 6 slots and 27 items at the maximum.