Geometric Batch Optimization for the Packing Equal Circles in a Circle Problem on Large Scale

TL;DR

This paper introduces a novel geometric batch optimization approach combined with heuristic search and adaptive neighbor maintenance to efficiently solve large-scale equal circle packing problems within reasonable computational times.

Contribution

It presents a new geometric batch optimization method and a heuristic search technique for large-scale circle packing, outperforming existing algorithms and reducing resource requirements.

Findings

Outperformed state-of-the-art algorithms on large scale instances

Found 95 improved solutions out of 101 large scale benchmark instances

Achieved significant speedup and memory reduction in optimization process

Abstract

The problem of packing equal circles in a circle is a classic and famous packing problem, which is well-studied in academia and has a variety of applications in industry. This problem is computationally challenging, and researchers mainly focus on small-scale instances with the number of circular items n less than 320 in the literature. In this work, we aim to solve this problem on large scale. Specifically, we propose a novel geometric batch optimization method that not only can significantly speed up the convergence process of continuous optimization but also reduce the memory requirement during the program's runtime. Then we propose a heuristic search method, called solution-space exploring and descent, that can discover a feasible solution efficiently on large scale. Besides, we propose an adaptive neighbor object maintenance method to maintain the neighbor structure applied in the continuous optimization process. In this way, we can find high-quality solutions on large scale instances within reasonable computational times. Extensive experiments on the benchmark instances sampled from n = 300 to 1,000 show that our proposed algorithm outperforms the state-of-the-art algorithms and performs excellently on large scale instances. In particular, our algorithm found 10 improved solutions out of the 21 well-studied moderate scale instances and 95 improved solutions out of the 101 sampled large scale instances. Furthermore, our geometric batch optimization, heuristic search, and adaptive maintenance methods are general and can be adapted to other packing and continuous optimization problems.