A Fast and Effective Breakpoints Heuristic Algorithm for the Quadratic Knapsack Problem

TL;DR

The paper introduces QKBP, a fast heuristic algorithm for the Quadratic Knapsack Problem that efficiently generates high-quality solutions across large instances by leveraging breakpoints and a greedy heuristic.

Contribution

It presents a novel, scalable heuristic algorithm for QKP based on breakpoints and a new compact problem formulation, outperforming existing methods in speed and solution quality.

Findings

QKBP consistently produces high-quality solutions across various instances.

QKBP is significantly faster than existing leading algorithms.

The algorithm scales well to instances with up to 10,000 elements.

Abstract

The Quadratic Knapsack Problem (QKP) involves selecting a subset of elements that maximizes the sum of pairwise and singleton utilities without exceeding a given budget. The pairwise utilities are nonnegative, the singleton utilities may be positive, negative, or zero, and the node costs are nonnegative. We introduce a Breakpoints Algorithm for QKP, named QKBP, which is based on a technique proposed in \cite{Hoc09} for efficiently generating the concave envelope of the solutions to the relaxation of the problem for all values of the budget. Our approach utilizes the fact that breakpoints in the concave envelopes are optimal solutions for their respective budgets. For budgets between breakpoints, a fast greedy heuristic derives high-quality solutions from the optimal solutions of adjacent breakpoints. The QKBP algorithm is a heuristic which is highly scalable due to an efficient parametric cut procedure used to generate the concave envelope. This efficiency is further improved by a newly developed compact problem formulation. Our extensive computational study on both existing and new benchmark instances, with up to 10,000 elements, shows that while some leading algorithms perform well on a few instances, QKBP consistently delivers high-quality solutions regardless of instance size, density, or budget. Moreover, QKBP achieves these results in significantly faster running times than all leading algorithms. The source code of the QKBP algorithm, the benchmark instances, and the detailed results are publicly available on GitHub.