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Dynamic combinatorial optimisation problems: an analysis of the subset sum problem

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Abstract

The field of evolutionary computation has traditionally focused on static optimisation problems. Recently, many new approaches have been proposed that adapt traditional evolutionary algorithms to the dynamic domain to deal with the task of tracking high-quality solutions as the search space changes over time. These novel algorithms are subsequently evaluated on a wide range of different optimisation problems, including well-specified benchmark generators. However, due to a lack of theoretical results, as well as a general lack of references to actual real-world scenarios, it is not entirely clear whether these benchmarks capture any of the characteristics found in NP-hard dynamic optimisation problems. In this paper, we extensively analyse the properties of the NP-hard (dynamic) subset sum problem. In particular, we highlight the correlation between the dynamic parameters of the problem and the resulting movement of the global optimum. It is shown by empirical means that the degree to which the global optimum moves in response to the underlying dynamics is correlated only in specific cases. Furthermore, the role of the representation used to encode the problem, as well as the impact of the formulation of the objective function on the dynamics are also discussed.

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Notes

  1. The update period, for instance, is often set to correspond to multiples of the population size of the algorithm considered.

  2. Here we use the terms static, stationary and time invariant interchangeably to refer to problems that do not change over time. Problems that do change over time, on the other hand, are described as dynamic or time variant.

  3. Assuming it is computationally tractable to detect when a change has occurred.

  4. This approach has actually been suggested to create dynamic benchmark problems in the combinatorial domain with controllable distances; see (Younes et al. 2005).

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Acknowledgments

This work was supported by EPSRC Grant no. EP/E058884/1 entitled “Evolutionary Algorithms for Dynamic Optimisation Problems: Design, Analysis and Applications”. The authors are grateful for useful discussions with Trung Thanh Nguyen regarding this work.

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  1. School of Computer Science, University of Birmingham, Birmingham, B15 2TT, UK

    Philipp Rohlfshagen & Xin Yao

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  1. Philipp Rohlfshagen
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  2. Xin Yao
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Correspondence to Philipp Rohlfshagen.

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Rohlfshagen, P., Yao, X. Dynamic combinatorial optimisation problems: an analysis of the subset sum problem. Soft Comput 15, 1723–1734 (2011). https://doi.org/10.1007/s00500-010-0616-9

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