Comparing Communities of Optima with Funnels in Combinatorial Fitness Landscapes

The existence of sub-optimal funnels in combinatorial fitness landscapes has been linked to search difficulty. The exact nature of these structures — and how commonly they appear — is not yet fully understood. Improving our understanding of funnels could help with designing effective diversification mechanisms for a ‘smoothing’ effect, making optimisation easier. We model fitness landscapes as local optima networks. The relationship between communities of local optima found by network clustering algorithms and funnels is explored. Funnels are identified using the notion of monotonic sequences from the study of energy landscapes in theoretical chemistry. NK Landscapes and the Quadratic Assignment Problem are used as case studies. Our results show that communities are linked to funnels. The analysis exhibits relationships between these landscape structures and the performance of trajectory-based metaheuristics such as Simulated Annealing (SA) and Iterated Local Search (ILS). In particular, ILS gets trapped in funnels, and modular communities of optima slow it down. The funnels contribute to lower success for SA. We show that increasing the strength of ILS perturbation helps to ‘smooth’ the funnels and improves performance in multi-funnel landscapes.Authors listed as ECOM Trac

Communities of Local Optima as Funnels in Fitness Landscapes

We conduct an analysis of local optima networks extracted from fitness landscapes of the Kauffman NK model under iterated local search. Applying the Markov Cluster Algorithm for community detection to the local optima networks, we find that the landscapes consist of multiple clusters. This result complements recent findings in the literature that landscapes often decompose into multiple funnels, which increases their difficulty for iterated local search. Our results suggest that the number of clusters as well as the size of the cluster in which the global optimum is located are correlated to the search difficulty of landscapes. We conclude that clusters found by community detection in local optima networks offer a new way to characterize the multi-funnel structure of fitness landscapes

Additional Dimensions to the Study of Funnels in Combinatorial Landscapes

The global structure of travelling salesman's fitness landscapes has recently revealed the presence of multiple `funnels'. This implies that local optima are organised into several clusters, so that a particular local optimum largely belongs to a particular funnel. Such a global structure can increase search difficulty, especially, when the global optimum is located in a deep, narrow funnel. Our study brings more precision (and dimensions) to the notion of funnels with a data-driven approach using Local Optima Networks and the Chained Lin-Kernighan heuristic. We start by exploring the funnel 'floors', characterising them using the notion of communities from complex networks. We then analyse the more complex funnel 'basins'. Since their depth is relevant to search, we visualise them in 3D. Our study, across a set of TSP instances, reveals a multi-funnel structure in most of them. However, the specific topology varies across instances and relates to search difficulty. Finally, including a stronger perturbation into Chained Lin-Kernighan proved to smooth the funnel structure, reducing the number of funnels and enlarging the valley leading to global optima

The Effect of Landscape Funnels in QAPLIB Instances

The effectiveness of common metaheuristics on combinatorial optimisation problems can be limited by certain characteristics of the fitness landscape. We use the local optima network model to compress the ‘inherent structure’ of a problem space into a network whose structure relates to the empirical hardness of the underlying landscape. Monotonic sequences are used on the local optima networks of a benchmark set of QAP instances (QAPLIB) to expose landscape funnels. The results suggest links between features of these structures and lowered metaheuristic performance

On the Fractal Nature of Local Optima Networks

A Local Optima Network represents fitness landscape connectivity within the space of local optima as a mathematical graph. In certain other complex networks or graphs there have been recent observations made about inherent self-similarity. An object is said to be self-similar if it shows the same patterns when measured at different scales; another word used to convey self-similarity is fractal. The fractal dimension of an object captures how the detail observed changes with the scale at which it is measured, with a high fractal dimension being associated with complexity. We conduct a detailed study on the fractal nature of the local optima networks of a benchmark combinatorial optimisation problem (NK Landscapes). The results draw connections between fractal characteristics and performance by three prominent metaheuristics: Iterated Local Search, Simulated Annealing, and Tabu Search

Inferring Future Landscapes: Sampling the Local Optima Level

Connection patterns among Local Optima Networks (LONs) can inform heuristic design for optimisation. LON research has predominantly required complete enumeration of a fitness landscape, thereby restricting analysis to problems diminutive in size compared to real-life situations. LON sampling algorithms are therefore important. In this paper, we study LON construction algorithms for the Quadratic Assignment Problem (QAP). Using machine learning, we use estimated LON features to predict search performance for competitive heuristics used in the QAP domain. The results show that by using random forest regression, LON construction algorithms produce fitness landscape features which can explain almost all search variance. We find that LON samples better relate to search than enumerated LONs do. The importance of fitness levels of sampled LONs in search predictions is crystallised. Features from LONs produced by different algorithms are combined in predictions for the first time, with promising results for this ‘super-sampling’: a model to predict tabu search success explained 99% of variance. Arguments are made for the use-case of each LON algorithm and for combining the exploitative process of one with the exploratory optimisation of the other

Clarifying the Difference in Local Optima Network Sampling Algorithms

We conduct the first ever statistical comparison between two Local Optima Network (LON) sampling algorithms. These methodologies attempt to capture the connectivity in the local optima space of a fitness landscape. One sampling algorithm is based on a random-walk snowballing procedure, while the other is centred around multiple traced runs of an Iterated Local Search. Both of these are proposed for the Quadratic Assignment Problem (QAP), making this the focus of our study. It is important to note the sampling algorithm frameworks could easily be modified for other domains. In our study descriptive statistics for the obtained search space samples are contrasted and commented on. The LON features are also used in linear mixed models and random forest regression for predicting heuristic optimisation performance of two prominent heuristics for the QAP on the underlying combinatorial problems. The model results are then used to make deductions about the sampling algorithms’ utility. We also propose a specific set of LON metrics for use in future predictive models alongside previously-proposed network metrics, demonstrating the payoff in doing so

Visualising the Search Landscape of the Triangle Program

High order mutation analysis of a software engineering benchmark, including schema and local optima networks, suggests program improvements may not be as hard to find as is often assumed. 1) Bit-wise genetic building blocks are not deceptive and can lead to all global optima. 2) There are many neutral networks, plateaux and local optima, nevertheless in most cases near the human written C source code there are hill climbing routes including neutral moves to solutions

Towards population-based fitness landscape analysis using local optima networks

This is the author accepted manuscript. The final version is available from ACM via the DOI in this recordA fitness landscape describes the interaction of a search domain, a cost function on designs drawn from the domain, and a neighbourhood function defining the adjacency of designs - - induced by the optimisation method used. Fitness landscapes can be represented in a compact form as Local Optima Networks (LONs). Although research has been conducted on LONs in continuous domains, the majority of work has focused on combinatorial landscapes. LONs are often used to understand the landscape encountered by population-based search heuristics, but are usually constructed via point-based search. This paper proposes the first construction of LONs by a population-based algorithm, applied to continuous optimisation problems. We construct LONs for three benchmark functions with well-known global structure using the widely used Nelder-Mead downhill simplex algorithm, and contrast these to the LONs from a point-based approach. We also investigate the sensitivity of the LON visualisation to the downhill simplex algorithm's hyperparameters, by varying the initial step size of the simplex and the step size for connectivity of optima. Our results suggest that large initial simplex sizes fragment the landscape structure, and exclude some local optima from the fitness landscape.Engineering and Physical Sciences Research Council (EPSRC