Fast r-flip move evaluations via closed-form formulae for Boolean quadratic programming problems with generalized upper bound constraints

Abstract

We consider the Boolean Quadratic Programming problem with Generalized Upper Bound constraints (BQP-GUB). This problem belongs to the NP-hard complexity class of problems and is a generalization of the Quadratic Semi-Assignment Problem (QSAP), which has applications in different areas such as production planning and image segmentation. Non-exact methods for solving BQP-GUB instances are commonly based on exploring solution neighborhoods, which consist of flip moves that choose an even number of binary variables and flip their values to the complementary value (from 1 to 0 or from 0 to 1). In order to find fairly good solutions, these methods must evaluate a large number of flip moves, and that can be time consuming. The best-known formulae used to evaluate flip moves for BQP-GUB take O(nr) time, where n is the number of variables and r is the number of flips. In this paper, we seek to improve the processing time of evaluating flip moves. We extend the results in the literature and prove two closed-form formulae for evaluating flip moves in O(mr) and O(r2) times, where m is the number of variables with values equal to one. Additionally, for experimental purposes, we prove a reduction from QSAP to BQP-GUB. We report on computational experiments with local search and Iterated Tabu Search algorithms using our formulae and those in the literature. Our results show that the implementations using one of our formulae achieved the best performance in all of the experiments.