The Multi-Funnel Structure of TSP Fitness Landscapes: A Visual Exploration

Abstract

Abstract. We use the Local Optima Network model to study the structure of symmetric TSP fitness landscapes. The 'big-valley' hypothesis holds that for TSP and other combinatorial problems, local optima are not randomly distributed, instead they tend to be clustered around the global optimum. However, a recent study has observed that, for solutions close in evaluation to the global optimum, this structure breaks down into multiple valleys, forming what has been called 'multiple funnels'. The multiple funnel concept implies that local optima are organised into clusters, so that a particular local optimum largely belongs to a particular funnel. Our study is the first to extract and visualise local optima networks for TSP and is based on a sampling methodology relying on the Chained Lin-Kernighan algorithm. We confirm the existence of multiple funnels on two selected TSP instances, finding additional funnels in a previously studied instance. Our results suggests that transitions among funnels are possible using operators such as 'double-bridge'. However, for consistently escaping sub-optimal funnels, more robust escaping mechanisms are required.

Modelling genetic improvement landscapes with local optima networks

“…Their number is indicative of the multi-funnel global structure of the landscapes [11], and may directly relate to the empirical problem hardness from the point of an iterated local search [10]. Among the four benchmark instances, the one with the lowest success rate also has multiple sub-optimal sinks.…”

Communities of Local Optima as Funnels in Fitness Landscapes

“…This so-called big valley hypothesis holds for a variety of optimization problems including the traveling salesman problem (TSP) [4]. Recent studies [10,24,23] extend the big valley hypothesis as they find that there is a structure of multiple funnels (instead of one single cluster) in the fitness landscape, which leads to a higher search difficulty for algorithms based on the principle of iterated local search (ILS), a search strategy that combines local search with perturbation steps [17]. Such global multi-funnel structures have been observed before in continuous optimization [16,18,14], however a more detailed characterization of funnels in combinatorial spaces like is still lacking.…”

“…GECCO '16, July 20 -24, 2016, Denver, CO, USA Ochoa et al [24] propose to characterize funnels in combinatorial spaces using local optima networks (LONs, [22]). The idea of LONs was inspired by the study of energy landscapes [28], and it was found that energy landscapes often have a structure of multiple funnels as well [19].…”

Tunnelling Crossover Networks for the Asymmetric TSP

“…An alternative definition of edges was later proposed to account for escape probabilities among optima, that is, probabilities to hop from a local optimum to another after a perturbation (large mutation) followed by local search [18]. Recently, sampling approaches have been developed using escape edges in order to model landscapes of realistic size [6,12,13,11]. In particular, work on the symmetric Travelling Salesman Problem has revealed intriguing landscape visualisations, providing compelling evidence of the existence of multiple valleys or clusters of local optima (also called funnels) on the studied instances [12,13].…”

“…Recently, sampling approaches have been developed using escape edges in order to model landscapes of realistic size [6,12,13,11]. In particular, work on the symmetric Travelling Salesman Problem has revealed intriguing landscape visualisations, providing compelling evidence of the existence of multiple valleys or clusters of local optima (also called funnels) on the studied instances [12,13]. Most local optima network models so far consider transitions based on perturbation operators.…”