A Study of NK Landscapes' Basins and Local Optima Networks

We propose a network characterization of combinatorial fitness landscapes by adapting the notion of inherent networks proposed for energy surfaces (Doye, 2002). We use the well-known family of $NK$ landscapes as an example. In our case the inherent network is the graph where the vertices are all the local maxima and edges mean basin adjacency between two maxima. We exhaustively extract such networks on representative small NK landscape instances, and show that they are 'small-worlds'. However, the maxima graphs are not random, since their clustering coefficients are much larger than those of corresponding random graphs. Furthermore, the degree distributions are close to exponential instead of Poissonian. We also describe the nature of the basins of attraction and their relationship with the local maxima network.Comment: best paper nominatio

Instances of combinatorial optimization problems: complexity and generation

138 p.La optimización combinatoria considera problemas donde el objetivo es hallar el punto que maximiza o minimiza una función y donde el espacio de búsqueda es nito o innito numerable. La resolución de estos problemas es de gran importancia, ya que aparecen de forma natural en diferentes ámbitos como el mundo de la ciencia y de la ingeniería, la industria o la gestión

Complex-network analysis of combinatorial spaces: The NK landscape case

We propose a network characterization of combinatorial fitness landscapes by adapting the notion of inherent networks proposed for energy surfaces. We use the well-known family of NK landscapes as an example. In our case the inherent network is the graph whose vertices represent the local maxima in the landscape, and the edges account for the transition probabilities between their corresponding basins of attraction. We exhaustively extracted such networks on representative NK landscape instances, and performed a statistical characterization of their properties. We found that most of these network properties are related to the search difficulty on the underlying NK landscapes with varying values of K.Comment: arXiv admin note: substantial text overlap with arXiv:0810.3492, arXiv:0810.348

A Large Population Size Can Be Unhelpful in Evolutionary Algorithms

The utilization of populations is one of the most important features of evolutionary algorithms (EAs). There have been many studies analyzing the impact of different population sizes on the performance of EAs. However, most of such studies are based computational experiments, except for a few cases. The common wisdom so far appears to be that a large population would increase the population diversity and thus help an EA. Indeed, increasing the population size has been a commonly used strategy in tuning an EA when it did not perform as well as expected for a given problem. He and Yao (2002) showed theoretically that for some problem instance classes, a population can help to reduce the runtime of an EA from exponential to polynomial time. This paper analyzes the role of population further in EAs and shows rigorously that large populations may not always be useful. Conditions, under which large populations can be harmful, are discussed in this paper. Although the theoretical analysis was carried out on one multi-modal problem using a specific type of EAs, it has much wider implications. The analysis has revealed certain problem characteristics, which can be either the problem considered here or other problems, that lead to the disadvantages of large population sizes. The analytical approach developed in this paper can also be applied to analyzing EAs on other problems.Comment: 25 pages, 1 figur

Hyperspace geography: Visualizing fitness landscapes beyond 4D

Human perception is finely tuned to extract structure about the 4D world of time and space as well as properties such as color and texture. Developing intuitions about spatial structure beyond 4D requires exploiting other perceptual and cognitive abilities. One of the most natural ways to explore complex spaces is for a user to actively navigate through them, using local explorations and global summaries to develop intuitions about structure, and then testing the developing ideas by further exploration. This article provides a brief overview of a technique for visualizing surfaces defined over moderate-dimensional binary spaces, by recursively unfolding them onto a 2D hypergraph. We briefly summarize the uses of a freely available Web-based visualization tool, Hyperspace Graph Paper (HSGP), for exploring fitness landscapes and search algorithms in evolutionary computation. HSGP provides a way for a user to actively explore a landscape, from simple tasks such as mapping the neighborhood structure of different points, to seeing global properties such as the size and distribution of basins of attraction or how different search algorithms interact with landscape structure. It has been most useful for exploring recursive and repetitive landscapes, and its strength is that it allows intuitions to be developed through active navigation by the user, and exploits the visual system's ability to detect pattern and texture. The technique is most effective when applied to continuous functions over Boolean variables using 4 to 16 dimensions

On the Runtime Analysis of the Clearing Diversity-Preserving Mechanism

Clearing is a niching method inspired by the principle of assigning the available resources among a niche to a single individual. The clearing procedure supplies these resources only to the best individual of each niche: the winner. So far, its analysis has been focused on experimental approaches that have shown that clearing is a powerful diversity-preserving mechanism. Using rigorous runtime analysis to explain how and why it is a powerful method, we prove that a mutation-based evolutionary algorithm with a large enough population size, and a phenotypic distance function always succeeds in optimising all functions of unitation for small niches in polynomial time, while a genotypic distance function requires exponential time. Finally, we prove that with phenotypic and genotypic distances clearing is able to find both optima for Twomax and several general classes of bimodal functions in polynomial expected time. We use empirical analysis to highlight some of the characteristics that makes it a useful mechanism and to support the theoretical results