The quadratic balanced optimization problem

TL;DR

This paper introduces the quadratic balanced optimization problem (QBOP) for equitable resource distribution with pairwise interactions, discusses its computational complexity, and presents algorithms with experimental validation.

Contribution

It defines QBOP, explores its complexity, and develops both exact and heuristic algorithms, including polynomially solvable cases for structured cost matrices.

Findings

QBOP is strongly NP-hard in general.

The paper presents effective exact and heuristic algorithms.

Polynomial solutions exist for certain structured cost matrices.

Abstract

We introduce the quadratic balanced optimization problem (QBOP) which can be used to model equitable distribution of resources with pairwise interaction. QBOP is strongly NP-hard even if the family of feasible solutions has a very simple structure. Several general purpose exact and heuristic algorithms are presented. Results of extensive computational experiments are reported using randomly generated quadratic knapsack problems as the test bed. These results illustrate the efficacy of our exact and heuristic algorithms. We also show that when the cost matrix is specially structured, QBOP can be solved as a sequence of linear balanced optimization problems. As a consequence, we have several polynomially solvable cases of QBOP.