Constructing the Neighborhood Structure of VNS Based on Binomial Distribution for Solving QUBO Problems

Abstract

The quadratic unconstrained binary optimization (QUBO) problem is categorized as an NP-hard combinatorial optimization problem. The variable neighborhood search (VNS) algorithm is one of the leading algorithms used to solve QUBO problems. As neighborhood structure change is the central concept in the VNS algorithm, the design of the neighborhood structure is crucial. This paper presents a modified VNS algorithm called “B-VNS”, which can be used to solve QUBO problems. A binomial trial was used to construct the neighborhood structure, and this was used with the aim of reducing computation time. The B-VNS and VNS algorithms were tested on standard QUBO problems from Glover and Beasley, on standard max-cut problems from Helmberg–Rendl, and on those proposed by Burer, Monteiro, and Zhang. Finally, Mann–Whitney tests were conducted using α=0.05, to statistically compare the performance of the two algorithms. It was shown that the B-VNS and VNS algorithms are able to provide good solutions, but the B-VNS algorithm runs substantially faster. Furthermore, the B-VNS algorithm performed the best in all of the max-cut problems, regardless of problem size, and it performed the best in QUBO problems, with sizes less than 500. The results suggest that the use of binomial distribution, to construct the neighborhood structure, has the potential for further development.