A Greedy, Generative, Lattice Representation for Point Packing

Abstract

Point packings in the unit square are placements of n points in the unit square that maximize the minimum distance between any two of the points. Such packings are surrogates for the 2D-stock cutting problem. In this study we examine a greedy generative representation for the point packing problem and extend the problem to higher dimensions. This representation uses a greedy algorithm to select points generated as whole-number linear combinations of vectors. This means that sets of vectors are evolved. The lattice generated by the vectors, taken modulo one, yields a set of points that can be greedily filtered to a dense point packing. The focus of evolution is the choice of vector generators for the lattice.A parameter study is performed comparing two mutation operators, different rates of application for mutation, and different population sizes. The generative representation is found to efficiently locate large point packings, while using relatively few real-valued parameters. The number of real parameters used to specify a point packing may be chosen. This novel control value is shown to have a substantial impact on results. A preliminary application of point packings, as population initializers, is demonstrated. Using a point packing as an initial population for an evolutionary optimizer can yield improved performance by providing more even sampling of the optimization domain and, in this study, it is shown that use of a point packing improves performance in a higher dimensional test problem, but not in a lower dimensional one.