Spin-Variable Reduction Method for Handling Linear Equality Constraints in Lsing Machines

Research output: Contribution to journalArticlepeer-review


We propose a spin-variable reduction method for Ising machines to handle linear equality constraints in a combinatorial optimization problem. Ising machines including quantum-annealing machines can effectively solve combinatorial optimization problems. They are designed to find the lowest-energy solution of a quadratic unconstrained binary optimization (QUBO), which is mapped from the combinatorial optimization problem. The proposed method reduces the number of binary variables to formulate the QUBO compared to the conventional penalty method. We demonstrate a sufficient condition to obtain the optimum of the combinatorial optimization problem in the spin-variable reduction method and its general applicability. We apply it to typical combinatorial optimization problems, such as the graph κ-partitioning problem and the quadratic assignment problem. Experiments using simulated-annealing and quantum-annealing based Ising machines demonstrate that the spin-variable reduction method outperforms the penalty method. The proposed method extends the application of Ising machines to larger-size combinatorial optimization problems with linear equality constraints.

Original languageEnglish
Pages (from-to)1-14
Number of pages14
JournalIEEE Transactions on Computers
Publication statusAccepted/In press - 2023


  • Combinatorial optimization problem
  • Ising machine
  • Ising model
  • Linear programming
  • Metaheuristics
  • Optimization
  • Quantum annealing
  • Quantum annealing
  • Search problems
  • Simulated annealing
  • Simulated annealing
  • Stationary state
  • Symbols
  • Variable reduction

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Hardware and Architecture
  • Computational Theory and Mathematics


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