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

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Evolutionary Computation in Combinatorial Optimization (EvoCOP 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10197))

Abstract

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.

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Acknowledgements

This work was supported by the Leverhulme Trust [award number RPG-2015-395] and 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 Stirling Online Repository for Research Data (http://hdl.handle.net/11667/85).

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Authors and Affiliations

  1. Computing Science and Mathematics, University of Stirling, Stirling, Scotland, UK

    Gabriela Ochoa, Nadarajen Veerapen & Fabio Daolio

  2. Faculty of Business and Economics, Information Systems Department, University of Lausanne, Lausanne, Switzerland

    Marco Tomassini

Authors
  1. Gabriela Ochoa
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  2. Nadarajen Veerapen
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  3. Fabio Daolio
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  4. Marco Tomassini
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Corresponding author

Correspondence to Gabriela Ochoa .

Editor information

Editors and Affiliations

  1. Austrian Institute of Technology, Vienna, Austria

    Bin Hu

  2. University of Manchester, Manchester, United Kingdom

    Manuel López-Ibáñez

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Ochoa, G., Veerapen, N., Daolio, F., Tomassini, M. (2017). Understanding Phase Transitions with Local Optima Networks: Number Partitioning as a Case Study. In: Hu, B., López-Ibáñez, M. (eds) Evolutionary Computation in Combinatorial Optimization. EvoCOP 2017. Lecture Notes in Computer Science(), vol 10197. Springer, Cham. https://doi.org/10.1007/978-3-319-55453-2_16

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  • DOI: https://doi.org/10.1007/978-3-319-55453-2_16

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