An efficient quasi-physical quasi-human algorithm for packing equal circles in a circular container

This paper addresses the equal circle packing problem, and proposes an efficient Quasi-physical Quasi-human (QPQH) algorithm. QPQH is based on a modified Broyden-Fletcher-GoldfarbShanno (BFGS) algorithm which we call the local BFGS and a new basin hopping strategy based on a Chinese idiom: alternate tension with relaxation. Starting from a random initial layout, we apply the local BFGS algorithm to reach a local minimum layout. The local BFGS algorithm fully utilizes the neighborhood information of each circle to considerably speed up the running time of the gradient descent process, and the efficiency is very apparent for large scale instances. When yielding a local minimum layout, the new basin-hopping strategy is to shrink the container size to different extent to generate several new layouts. Experimental results indicate that the new basin-hopping strategy is very efficient, especially for a type of layout with comparatively dense packing in the center and comparatively sparse packing around the boundary of the container. We test QPQH on the instances of n = 1, 2, · · · , 320, and obtain 66 new layouts which have smaller container sizes than the current best-known results reported in literature.