Cover-Encodings of Fitness Landscapes

The traditional way of tackling discrete optimization problems is by using local search on suitably defined cost or fitness landscapes. Such approaches are however limited by the slowing down that occurs when the local minima that are a feature of the typically rugged landscapes encountered arrest the progress of the search process. Another way of tackling optimization problems is by the use of heuristic approximations to estimate a global cost minimum. Here we present a combination of these two approaches by using cover-encoding maps which map processes from a larger search space to subsets of the original search space. The key idea is to construct cover-encoding maps with the help of suitable heuristics that single out near-optimal solutions and result in landscapes on the larger search space that no longer exhibit trapping local minima. We present cover-encoding maps for the problems of the traveling salesman, number partitioning, maximum matching and maximum clique; the practical feasibility of our method is demonstrated by simulations of adaptive walks on the corresponding encoded landscapes which find the global minima for these problems.Comment: 15 pages, 4 figure

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

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

Landscape Encodings Enhance Optimization

Hard combinatorial optimization problems deal with the search for the minimum cost solutions (ground states) of discrete systems under strong constraints. A transformation of state variables may enhance computational tractability. It has been argued that these state encodings are to be chosen invertible to retain the original size of the state space. Here we show how redundant non-invertible encodings enhance optimization by enriching the density of low-energy states. In addition, smooth landscapes may be established on encoded state spaces to guide local search dynamics towards the ground state

How Perturbation Strength Shapes the Global Structure of TSP Fitness Landscapes

Local optima networks are a valuable tool used to analyse and visualise the global structure of combinatorial search spaces; in particular, the existence and distribution of multiple funnels in the landscape. We extract and analyse the networks induced by Chained-LK, a powerful iterated local search for the TSP, on a large set of randomly generated (Uniform and Clustered) instances. Results indicate that increasing the perturbation strength employed by Chained-LK modifies the landscape's global structure, with the effect being markedly different for the two classes of instances. Our quantitative analysis shows that several funnel metrics have stronger correlations with Chained-LK success rate than the number of local optima, indicating that global structure clearly impacts search performance

Perturbation strength and the global structure of qap fitness landscapes

We study the effect of increasing the perturbation strength on the global structure of QAP fitness landscapes induced by Iterated Local Search (ILS). The global structure is captured with Local Optima Networks. Our analysis concentrates on the number, characteristics and distribution of funnels in the landscape, and how they change with increasing perturbation strengths. Well-known QAP instance types are considered. Our results confirm the multi-funnel structure of QAP fitness landscapes and clearly explain, visually and quantitatively, why ILS with large perturbation strengths produces better results. Moreover, we found striking differences between randomly generated and real-world instances, which warns about using synthetic benchmarks for (manual or automatic) algorithm design and tuning