Bit duplication technique to generate hard QUBO problems

Abstract

Quadratic 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 paper is to present the bit duplication technique, which can generate a hard QUBO problem from any QUBO problem so that their optimal solutions take the same energy value. The idea of the bit duplication technique is to duplicate bits and then to give constraints so that the corresponding two bits take the same binary values. By this technique, any QUBO problem with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> bits is converted to a hard QUBO problem with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$2n$</tex> bits. We use random QUBO problems and N-Queen QUBO problems for experiments. The performance of QUBO solvers including Gurobi optimizer, Fixstars Amplify AE, OpenJij with SA, D-Wave Hybrid solver, and ABS QUBO solver are evaluated for solving these QUBO problems. The experimental results show that generated QUBO problems by the bit duplication technique are much harder than the original QUBO problems. 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.