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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10197)


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.



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 (


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Computing Science and MathematicsUniversity of StirlingStirlingUK
  2. 2.Faculty of Business and Economics, Information Systems DepartmentUniversity of LausanneLausanneSwitzerland

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