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

Understanding Phase Transitions with Local Optima Networks: Number Partitioning as a Case Study

Phase transitions play an important role in understanding search difficulty in combinatorial optimisation. However, previous attempts have not revealed a clear link between fitness landscape properties and the phase transition. We explore whether the global landscape structure of the number partitioning problem changes with the phase transition. Using the local optima network model, we analyse a number of instances before, during, and after the phase transition. We compute relevant network and neutrality metrics; and importantly, identify and visualise the funnel structure with an approach (monotonic sequences) inspired by theoretical chemistry. While most metrics remain oblivious to the phase transition, our results reveal that the funnel structure clearly changes. Easy instances feature a single or a small number of dominant funnels leading to global optima; hard instances have a large number of suboptimal funnels attracting the search. Our study brings new insights and tools to the study of phase transitions in combinatorial optimisation

Official Discrepancies: Kosovo Independence and Western European Rhetoric

This article examines approaches and official discrepancies characterising Western European rhetoric with regard to the Kosovo status question. Since the early 1980s, Kosovo has been increasingly present in European debates, culminating with the 1999 international intervention in the region and subsequent talks about its final status. Although the Kosovo Albanians proclaimed independence in February 2008 and the majority of EU Member States decided to recognise Kosovo as an independent state, Western European rhetoric has been rather divided. This article shows that in addition to five EU members who have decided not to recognise Kosovo from the very beginning, and thus are powerful enough to affect its further progress, both locally and internationally, some of the recognisers, although having abandoned the policy of ‘standards before status’, have also struggled to develop full support for the province – a discrepancy that surely questions the overall Western support for Kosovo’s independence

Differential evolution and non-separability: using selective pressure to focus search

Recent results show that the Differential Evolution algorithm has significant difficulty on functions that are not linearly separable. On such functions, the algorithm must rely primarily on its differential mutation procedure which, unlike its recombination strategy, is rotationally invariant. We conjecture that this mutation strategy lacks sufficient selective pressure when appointing parent and donor vectors to have satisfactory exploitative power on non-separable functions. We find that imposing pressure in the form of rank-based differential mutation results in a significant improvement of exploitation on rotated benchmarks.Andrew M. Sutton, Monte Lunacek, L. Darrell Whitleyhttp://www.sigevo.org/gecco-2007/index.htm

The impact of global structure on search

Also published as book chapter: Parallel Problem Solving from Nature – PPSN X, 2008 / Günter Rudolph, Thomas Jansen, Simon Lucas, Carlo Poloni, Nicola Beume (eds.), pp.498-507Population-based methods are often considered superior on multimodal functions because they tend to explore more of the fitness landscape before they converge. We show that the effectiveness of this strategy is highly dependent on a function’s global structure. When the local optima are not structured in a predictable way, exploration can misguide search into sub-optimal regions. Limiting exploration can result in a better non-intuitive global search strategy.Monte Lunacek, Darrell Whitley, and Andrew Sutto