An Efficient Quasi-physical Algorithm for Packing Equal Circles in a Circular Container.

This paper addresses the equal circle packing problem, and proposes an efficient quasi-physical algorithm(EQPA). EQPA is based on an improved BFGS algorithm and a new basin hopping strategy. Starting form a random initial pattern, we use the modified BFGS algorithm to reach a local minimum pattern. The modified BFGS algorithm fully utilizes the neighborhood information and considerably speeds up the running time of the gradient descent process, and the efficiency is more apparent for larger scale instances. The new basin-hopping strategy is to shrink the container size when yielding a local minimum pattern. Experimental results indicate that the new basin-hopping strategy is very efficient, especially for a type of pattern with comparatively dense packing in the center and sparse packing around the bounding. We test EQPA on the instances of $n = 1,2,cdots,320$, and obtain 66 new patterns which have smaller container sizes than the current best-known results reported in literature.