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.
['Gabriela Ochoa', 'Nadarajen Veerapen']
Additional Dimensions to the Study of Funnels in Combinatorial Landscapes