Perturbation-Based Thresholding Search for Packing Equal Circles and Spheres

Abstract

This paper presents an effective perturbation-based thresholding search for two popular and challenging packing problems with minimal containers: packing N identical circles in a square and packing N identical spheres in a cube. Following the penalty function approach, we handle these constrained optimization problems by solving a series of unconstrained optimization subproblems with fixed containers. The proposed algorithm relies on a two-phase search strategy that combines a thresholding search method reinforced by two general-purpose perturbation operators and a container adjustment method. The performance of the algorithm is assessed relative to a large number of benchmark instances widely studied in the literature. Computational results show a high performance of the algorithm on both problems compared with the state-of-the-art results. For circle packing, the algorithm improves 156 best-known results (new upper bounds) in the range of [Formula: see text] and matches 242 other best-known results. For sphere packing, the algorithm improves 66 best-known results in the range of [Formula: see text], whereas matching the best-known results for 124 other instances. Experimental analyses are conducted to shed light on the main search ingredients of the proposed algorithm consisting of the two-phase search strategy, the mixed perturbation and the parameters. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was supported by the National Natural Science Foundation of China [Grants 61703213 and 61933005]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.1290 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0004 ) at ( http://dx.doi.org/10.5281/zenodo.7579558 ).