How Partition Crossover exposes parallel lattices and the fractal structure of k-bounded functions

Abstract

A combination of recombination and local search can expose the existence of an exponential number of parallel lattices that span the search space for all classes of k-bounded pseudo-Boolean functions, including MAX-kSAT problems. These "parallel" lattices sometimes have identical evaluations shifted by a constant. We use Partition Crossover to aid in the discovery of lattices, which are sets of 2q possible offspring from recombination events, organized into q-dimensional hypercubes, where q is the number of recombining components given two parents. Finally, we show that recursively embedded subspace lattices display a fractal structure, which can be captured using rewrite rules based on a Lindenmayer system that accurately model how local optima are distributed across different size lattices.