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arXiv · 2604.03078

The Quadratic Bin Packing Problem: Exact Formulations and Algorithm

Vítor Gomes Chagas·Alberto Locatelli·Flávio Keidi Miyazawa·Manuel Iori
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Abstract

In this article, we introduce and study the Quadratic Bin Packing Problem (QBPP), which generalizes the classical bin packing problem by introducing a fixed cost for each used bin and a pairwise cost (or profit) incurred whenever two items are packed together. Beyond its theoretical relevance, the QBPP is of practical interest due to its numerous real-world applications, mainly related to cluster analysis. To address the QBPP, we propose three compact mixed-integer linear programming (MILP) formulations, along with a set-partitioning formulation. For each compact model, we present an enhanced version with a strengthened continuous relaxation, while, for the set-partitioning formulation, we develop a tailored Branch-and-Price algorithm. Computational experiments on benchmark instances demonstrated that, while the enhanced compact formulations can be effectively solved by a standard MILP solver for small-sized instances, the Branch-and-Price approach delivered superior performance overall, especially on larger and more challenging instances.

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Vítor Gomes Chagas, Alberto Locatelli, Flávio Keidi Miyazawa, Manuel Iori. 2026-04-03. The Quadratic Bin Packing Problem: Exact Formulations and Algorithm. https://arxiv.org/abs/2604.03078

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