Bit duplication technique to generate hard quadratic unconstrained binary optimization problems with adjustable sizes

Abstract

SummaryQuadratic unconstrained binary optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The main contribution of this article is to propose the bit duplication technique that can specify the number of duplicated bits, so that it can generate hard QUBO problem with adjustable sizes. The idea is to duplicate specified number of bits and then to give constraints so that the corresponding two bits take the same binary values. By this technique, any QUBO problem with bits is converted to a hard QUBO problem with bits . We use random QUBO problems, N‐Queen problems, traveling salesman problem and maximum weight matching problems for experiments. The performance of QUBO solvers including Gurobi optimizer, Fixstars Amplify AE, OpenJij with SA, D‐Wave samplers with SA, D‐Wave hybrid and ABS2 QUBO solver are evaluated for solving these QUBO problems. The experimental results show that only a small scale of duplicated bits can make QUBO problems harder. Hence, the bit duplication technique is a potent method to generate hard QUBO problems and generated QUBO problems can be used as benchmark problems for evaluating the search performance of QUBO solvers.