Local Optima Networks of NK Landscapes with Neutrality

In previous work we have introduced a network-based model that abstracts many details of the underlying landscape and compresses the landscape information into a weighted, oriented graph which we call the local optima network. The vertices of this graph are the local optima of the given fitness landscape, while the arcs are transition probabilities between local optima basins. Here we extend this formalism to neutral fitness landscapes, which are common in difficult combinatorial search spaces. By using two known neutral variants of the NK family (i.e. NKp and NKq) in which the amount of neutrality can be tuned by a parameter, we show that our new definitions of the optima networks and the associated basins are consistent with the previous definitions for the non-neutral case. Moreover, our empirical study and statistical analysis show that the features of neutral landscapes interpolate smoothly between landscapes with maximum neutrality and non-neutral ones. We found some unknown structural differences between the two studied families of neutral landscapes. But overall, the network features studied confirmed that neutrality, in landscapes with percolating neutral networks, may enhance heuristic search. Our current methodology requires the exhaustive enumeration of the underlying search space. Therefore, sampling techniques should be developed before this analysis can have practical implications. We argue, however, that the proposed model offers a new perspective into the problem difficulty of combinatorial optimization problems and may inspire the design of more effective search heuristics.Comment: IEEE Transactions on Evolutionary Computation volume 14, 6 (2010) to appea

New insights on neutral binary representations for evolutionary optimization

This paper studies a family of redundant binary representations NNg(l, k), which are based on the mathematical formulation of error control codes, in particular, on linear block codes, which are used to add redundancy and neutrality to the representations. The analysis of the properties of uniformity, connectivity, synonymity, locality and topology of the NNg(l, k) representations is presented, as well as the way an (1+1)-ES can be modeled using Markov chains and applied to NK fitness landscapes with adjacent neighborhood.The results show that it is possible to design synonymously redundant representations that allow an increase of the connectivity between phenotypes. For easy problems, synonymously NNg(l, k) representations, with high locality, and where it is not necessary to present high values of connectivity are the most suitable for an efficient evolutionary search. On the contrary, for difficult problems, NNg(l, k) representations with low locality, which present connectivity between intermediate to high and with intermediate values of synonymity are the best ones. These results allow to conclude that NNg(l, k) representations with better performance in NK fitness landscapes with adjacent neighborhood do not exhibit extreme values of any of the properties commonly considered in the literature of evolutionary computation. This conclusion is contrary to what one would expect when taking into account the literature recommendations. This may help understand the current difficulty to formulate redundant representations, which are proven to be successful in evolutionary computation. (C) 2016 Elsevier B.V. All rights reserved

Adaptive walks on time-dependent fitness landscapes

The idea of adaptive walks on fitness landscapes as a means of studying evolutionary processes on large time scales is extended to fitness landscapes that are slowly changing over time. The influence of ruggedness and of the amount of static fitness contributions are investigated for model landscapes derived from Kauffman's $NK$ landscapes. Depending on the amount of static fitness contributions in the landscape, the evolutionary dynamics can be divided into a percolating and a non-percolating phase. In the percolating phase, the walker performs a random walk over the regions of the landscape with high fitness.Comment: 7 pages, 6 eps-figures, RevTeX, submitted to Phys. Rev.

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

Scuba Search : when selection meets innovation

We proposed a new search heuristic using the scuba diving metaphor. This approach is based on the concept of evolvability and tends to exploit neutrality in fitness landscape. Despite the fact that natural evolution does not directly select for evolvability, the basic idea behind the scuba search heuristic is to explicitly push the evolvability to increase. The search process switches between two phases: Conquest-of-the-Waters and Invasion-of-the-Land. A comparative study of the new algorithm and standard local search heuristics on the NKq-landscapes has shown advantage and limit of the scuba search. To enlighten qualitative differences between neutral search processes, the space is changed into a connected graph to visualize the pathways that the search is likely to follow

Multi-layer local optima networks for the analysis of advanced local search-based algorithms

A Local Optima Network (LON) is a graph model that compresses the fitness landscape of a particular combinatorial optimization problem based on a specific neighborhood operator and a local search algorithm. Determining which and how landscape features affect the effectiveness of search algorithms is relevant for both predicting their performance and improving the design process. This paper proposes the concept of multi-layer LONs as well as a methodology to explore these models aiming at extracting metrics for fitness landscape analysis. Constructing such models, extracting and analyzing their metrics are the preliminary steps into the direction of extending the study on single neighborhood operator heuristics to more sophisticated ones that use multiple operators. Therefore, in the present paper we investigate a twolayer LON obtained from instances of a combinatorial problem using bitflip and swap operators. First, we enumerate instances of NK-landscape model and use the hill climbing heuristic to build the corresponding LONs. Then, using LON metrics, we analyze how efficiently the search might be when combining both strategies. The experiments show promising results and demonstrate the ability of multi-layer LONs to provide useful information that could be used for in metaheuristics based on multiple operators such as Variable Neighborhood Search.Comment: Accepted in GECCO202

Fitness landscape of the cellular automata majority problem: View from the Olympus

In this paper we study cellular automata (CAs) that perform the computational Majority task. This task is a good example of what the phenomenon of emergence in complex systems is. We take an interest in the reasons that make this particular fitness landscape a difficult one. The first goal is to study the landscape as such, and thus it is ideally independent from the actual heuristics used to search the space. However, a second goal is to understand the features a good search technique for this particular problem space should possess. We statistically quantify in various ways the degree of difficulty of searching this landscape. Due to neutrality, investigations based on sampling techniques on the whole landscape are difficult to conduct. So, we go exploring the landscape from the top. Although it has been proved that no CA can perform the task perfectly, several efficient CAs for this task have been found. Exploiting similarities between these CAs and symmetries in the landscape, we define the Olympus landscape which is regarded as the ''heavenly home'' of the best local optima known (blok). Then we measure several properties of this subspace. Although it is easier to find relevant CAs in this subspace than in the overall landscape, there are structural reasons that prevent a searcher from finding overfitted CAs in the Olympus. Finally, we study dynamics and performance of genetic algorithms on the Olympus in order to confirm our analysis and to find efficient CAs for the Majority problem with low computational cost

Clustering of Local Optima in Combinatorial Fitness Landscapes

Using the recently proposed model of combinatorial landscapes: local optima networks, we study the distribution of local optima in two classes of instances of the quadratic assignment problem. Our results indicate that the two problem instance classes give rise to very different configuration spaces. For the so-called real-like class, the optima networks possess a clear modular structure, while the networks belonging to the class of random uniform instances are less well partitionable into clusters. We briefly discuss the consequences of the findings for heuristically searching the corresponding problem spaces.Comment: Learning and Intelligent OptimizatioN Conference (LION 5), Rome : Italy (2011