Improving coalition structure search with an imperfect algorithm: analysis and evaluation results

Abstract

Optimal Coalition Structure Generation (CSG) is a significant research problem in multi-agent systems that remains difficult to solve. This problem has many important applications in transportation, eCommerce, distributed sensor networks and others. The CSG problem is NP-complete and finding the optimal result for n agents needs to check \(O (n^n)\) possible partitions. The ODP–IP algorithm (Michalak et al. in Artif Intell 230:14–50, 2016) achieves the current lowest worst-case time complexity of \(O (3^n)\). In the light of its high computational time complexity, we devise an Imperfect Dynamic Programming (ImDP) algorithm for the CSG problem with runtime \(O (n2^n)\) given n agents. Imperfect algorithm means that there are some contrived inputs for which the algorithm fails to give the optimal result. We benchmarked ImDP against ODP–IP and proved its efficiency. Experimental results confirmed that ImDP algorithm performance is better for several data distributions, and for some it improves dramatically ODP–IP. For example, given 27 agents, with ImDP for agent-based uniform distribution time gain is 91% (i.e. 49 min).

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Notes

  1. 1.

    Use of \(Merge_1\) function for coalitions of size 1, 2 and 3 is redundant. We prove this later in property 1.

  2. 2.

    The set difference \(\{\mathcal C\setminus a_s\}\) is defined as \(\{{\mathcal {C}}\setminus a_s\}=\{x:x\in {\mathcal {C}} \text{ and } x\notin a_s\}\). We use \(\mathcal U\) to denote coalition \(\{{\mathcal {C}}\setminus a_s\}\)

  3. 3.

    \(^a\) This line merges each component of \(\mathcal X\) with another component of \(\mathcal Y\) one at a time and leaves the other parts unchanged.

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Acknowledgements

The research presented in this article is funded by “Visvesvaraya Ph.D. Scheme for Electronics & IT”, Grant No: PhD-MLA/4(29)/2015-16.

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Correspondence to Narayan Changder.

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A preliminary version of this paper have been accepted in the Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19) Student Abstract and Poster Program (SA-19).

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Changder, N., Aknine, S. & Dutta, A. Improving coalition structure search with an imperfect algorithm: analysis and evaluation results. Artif Intell Rev 54, 397–425 (2021). https://doi.org/10.1007/s10462-020-09850-5

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Keywords

  • Coalition structure generation
  • Dynamic programming
  • Coalition formation
  • Imperfect algorithm