Exact Methods for Recursive Circle Packing

TL;DR

This paper introduces an exact method for solving the Recursive Circle Packing Problem using a Dantzig--Wolfe reformulation and column generation, achieving tight bounds and improved solutions over heuristics.

Contribution

It is the first dedicated approach for RCPP that provides strong dual bounds and global optimality through an innovative reformulation and column generation technique.

Findings

Computes tight dual bounds for RCPP.

Often finds primal solutions better than existing heuristics.

Demonstrates effectiveness on large test sets.

Abstract

Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This problem is known as the Recursive Circle Packing Problem (RCPP). We present the first dedicated method for solving RCPP that provides strong dual bounds based on an exact Dantzig--Wolfe reformulation of a nonconvex mixed-integer nonlinear programming formulation. The key idea of this reformulation is to break symmetry on each recursion level by enumerating one-level packings, i.e., packings of circles into other circles, and by dynamically generating packings of circles into rectangles. We use column generation techniques to design a "price-and-verify" algorithm that solves this reformulation to global optimality. Extensive computational experiments on a large test set show that our method not only computes tight dual bounds, but often produces primal solutions better than those computed by heuristics from the literature.