Linearization via Ordering Variables in Binary Optimization for Ising Machines

Ising machines are next-generation computers expected for efficiently sampling near-optimal solutions of combinatorial oprimization problems. Combinatorial optimization problems are modeled as quadratic unconstrained binary optimization (QUBO) problems to apply an Ising machine. However, current state-of-the-art Ising machines still often fail to output near-optimal solutions due to the complicated energy landscape of QUBO problems. Furthermore, physical implementation of Ising machines severely restricts the size of QUBO problems to be input as a result of limited hardware graph structures. In this study, we take a new approach to these challenges by injecting auxiliary penalties preserving the optimum, which reduces quadratic terms in QUBO objective functions. The process simultaneously simplifies the energy landscape of QUBO problems, allowing search for near-optimal solutions, and makes QUBO problems sparser, facilitating encoding into Ising machines with restriction on the hardware graph structure. We propose linearization via ordering variables of QUBO problems as an outcome of the approach. By applying the proposed method to synthetic QUBO instances and to multi-dimensional knapsack problems, we empirically validate the effects on enhancing minor embedding of QUBO problems and performance of Ising machines.