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

  • Gabriela OchoaEmail author
  • Nadarajen Veerapen
  • Darrell Whitley
  • Edmund K. Burke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9554)


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.


Global Optimum Local Optimum Bond Distance Travel Salesman Problem Fitness Level 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by the UK’s Engineering and Physical Sciences Research Council [grant number EP/J017515/1].

Data Access. All data generated during this research are openly available from the Zenodo repository (


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Gabriela Ochoa
    • 1
    Email author
  • Nadarajen Veerapen
    • 1
  • Darrell Whitley
    • 2
  • Edmund K. Burke
    • 1
  1. 1.Computing Science and MathematicsUniversity of StirlingStirlingScotland, UK
  2. 2.Department of Computer ScienceColorado State UniversityFort CollinsUSA

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