The component model for elementary landscapes and partial neighborhoods

Theoretical Computer Science, 545, (2014), pp. 59-75Local search algorithms exploit moves on an adjacency graph of the search space. An “elementary landscape” exists if the objective function f is an eigenfunction of the Laplacian of the graph induced by the neighborhood operator; this allows various statistics about the neighborhood to be computed in closed form. A new component based model makes it relatively simple to prove that certain types of landscapes are elementary. The traveling salesperson problem, weighted graph (vertex) coloring and the minimum graph bisection problem yield elementary landscapes under commonly used local search operators. The component model is then used to efficiently compute the mean objective function value over partial neighborhoods for these same problems. For a traveling salesperson problem over n cities, the 2-opt neighborhood can be decomposed into ⌊n/2−1⌋ partial neighborhoods. For graph coloring and the minimum graph bisection problem, partial neighborhoods can be used to focus search on those moves that are capable of producing a solution with a strictly improving objective function value.Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant number FA9550-08-1-0422

Structural coherence of problem and algorithm: an analysis for EDAs on all 2-bit and 3-bit problems.

Metaheuristics assume some kind of coherence between decision and objective spaces. Estimation of Distribution algorithms approach this by constructing an explicit probabilistic model of high fitness solutions, the structure of which is intended to reflect the structure of the problem. In this context, 'structure' means the dependencies or interactions between problem variables in a probabilistic graphical model. There are many approaches to discovering these dependencies, and existing work has already shown that often these approaches discover 'unnecessary' elements of structure - that is, elements which are not needed to correctly rank solutions. This work performs an exhaustive analysis of all 2 and 3 bit problems, grouped into classes based on mononotic invariance. It is shown in [1] that each class has a minimal Walsh structure that can be used to solve the problem. We compare the structure discovered by different structure learning approaches to the minimal Walsh structure for each class, with summaries of which interactions are (in)correctly identified. Our analysis reveals a large number of symmetries that may be used to simplify problem solving. We show that negative selection can result in improved coherence between discovered and necessary structure, and conclude with some directions for a general programme of study building on this work