Discretization-Based Solution Approaches for the Circle Packing Problem

TL;DR

This paper introduces a discretization-based iterative approach for solving the circle packing problem, focusing on minimizing the container size through integer linear programming and advanced optimization techniques.

Contribution

It presents a novel discretization and bisection algorithm for circle packing, incorporating solution space reduction and variable elimination, with comprehensive computational evaluation.

Findings

The proposed method outperforms existing heuristics in certain instances.

It achieves competitive results compared to BARON and Gurobi.

The approach effectively reduces solution space and tightens bounds.

Abstract

The problem of packing a set of circles into the smallest surrounding container is considered. This problem arises in different application areas such as automobile, textile, food, and chemical industries. The so-called circle packing problem can be cast as a nonconvex quadratically constrained program, and is difficult to solve in general. An iterative solution approach based on a bisection-type algorithm on the radius of the larger circle is provided. The present algorithm discretizes the container into small cells and solves two different integer linear programming formulations proposed for a restricted and a relaxed version of the original problem. The present algorithm is enhanced with solution space reduction, bound tightening and variable elimination techniques. Then, a computational study is performed to evaluate the performance of the algorithm. The present algorithm is compared with BARON and Gurobi that solve the original nonlinear formulation and heuristic methods from literature, and obtain promising results.