The Quadratic Bin Packing Problem: Exact Formulations and Algorithm

TL;DR

This paper introduces the Quadratic Bin Packing Problem (QBPP), proposing new formulations and algorithms, including a Branch-and-Price method, with computational results showing its effectiveness on benchmark instances.

Contribution

The paper presents three compact MILP formulations, a set-partitioning formulation, and a tailored Branch-and-Price algorithm for QBPP, advancing solution methods for this generalized bin packing problem.

Findings

Enhanced formulations solve small instances efficiently.

Branch-and-Price outperforms other methods on larger instances.

Computational experiments validate the effectiveness of the proposed approaches.

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.