Abstract
Ising machines are promising alternatives to solve combinatorial optimization problems, which search for their quasi-optimal solutions with high speed and high accuracy. However, the obtained solution much depends on the initial spin states, since the computation time is finite. Moreover, due to their probabilistic nature, they cannot always satisfy the constraints given to combinatorial optimization problems. In this paper, we propose a three-stage annealing method, targeting a slot-placement problem as a typical but difficult example of combinatorial optimization problems. The proposed method is composed of an initial process, an annealing process, and a correction process. The initial process and the correction process are executed by a classical computer while the annealing process is executed by an Ising machine. In the initial process, we give initial spin values that lead to a relatively good solution to the combinatorial optimization problem, which satisfies the given constraints. Then, the annealing process is executed by an Ising machine, and the solution obtained by the annealing process is further corrected to satisfy the constraints. The experimental results demonstrate that the proposed method reduces a minimum total weighted wiring length by 0.0898%–2.45% on average depending on the initial process methods used, compared to the existing method. The mean total weighted wiring length is reduced by 2.79%–7.08% on average depending on the initial process methods used.
Highlights
ORGANIZATION OF THIS PAPER This paper is organized as follows: Section II summarizes the related works; Section III defines the slot-placement problem and its constraints and objective function; Section IV introduces the Ising model and quadratic unconstrained binary optimization (QUBO) model and explains the slot-placement problem mapping to the QUBO model; Section V proposes a three-stage annealing method, where we propose three initial processes, i.e., a pair-wise exchange method, a random exchange method, and a clustergrowth method, and a correction process; Section VI demonstrates the effectiveness of the proposed method using the Ising machine hardware; and Section VII summarizes this paper and gives several concluding remarks
In any cases except for small examples, the total weighted wiring length is reduced by introducing the initial process, which definitely indicates that the three-stage annealing process is effective to the slot-placement problem
These results demonstrate that the annealing process successfully improves the total weighted wiring length, even after the initial solution is given by the initial process
Summary
A combinatorial optimization problem is a problem to find a combination of variables that maximizes or minimizes an objective function while satisfying its given constraints. In [16], after the quasi-optimal solutions are obtained by using an Ising machine targeting the slot-placement problem, they are improved as post-processing using a classical computer, so that they satisfy the slot-placement constraint. E. CONTRIBUTIONS OF THIS PAPER The contributions of this paper are summarized as follows: 1) We propose a three-stage annealing method solving the slot-placement problem, which efficiently utilizes an Ising machine and obtains a feasible quasi-optimal solution to the slot-placement problem. Compared to the two-stage annealing method [16] not including any initial process, the proposed method reduces the minimum total weighted wiring length by approximately 2.42% on average in the case of using the pairwise exchange method, approximately 2.45% on average in the case of using the random exchange method, and approximately 0.0898% on average in the case of using the cluster-growth method. The case of using the random exchange method, and approximately 2.79% on average in the case of using the cluster-growth method
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