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

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

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

References

  • Adams J et al (2010) Approximate coalition structure generation. In: Proceedings of the 24th AAAI conference on artificial intelligence, AAAI, pp 854–859

  • Bistaffa F, Farinelli A, Cerquides J, Rodríguez-Aguilar J, Ramchurn SD (2014) Anytime coalition structure generation on synergy graphs. In: Proceedings of the 2014 international conference on Autonomous agents and multi-agent systems. International Foundation for Autonomous Agents and Multiagent Systems, pp 13–20

  • Bistaffa F, Farinelli A, Cerquides J, Rodríguez-Aguilar J, Ramchurn SD (2017) Algorithms for graph-constrained coalition formation in the real world. ACM Trans Intell Syst Technol (TIST) 8(4):60

    Google Scholar 

  • Björklund A, Husfeldt T, Koivisto M (2009) Set partitioning via inclusion-exclusion. SIAM J Comput 39(2):546–563

    Article  MathSciNet  Google Scholar 

  • Cruz F, Espinosa A, Moure JC, Cerquides J, Rodriguez-Aguilar JA, Svensson K, Ramchurn SD (2017) Coalition structure generation problems: optimization and parallelization of the idp algorithm in multicore systems. Concur Comput Pract Exp 29(5):3969

    Article  Google Scholar 

  • Dang VD, Jennings NR (2004) Generating coalition structures with finite bound from the optimal guarantees. In: Proceedings of the third international joint conference on autonomous agents and multiagent systems, vol 2. IEEE Computer Society, pp 564–571

  • Dang VD, Dash RK, Rogers A, Jennings NR (2006) Overlapping coalition formation for efficient data fusion in multi-sensor networks. AAAI 6:635–640

    Google Scholar 

  • Di Mauro N, Basile TM, Ferilli S, Esposito F (2010) Coalition structure generation with grasp. In: International conference on artificial intelligence: methodology, systems, and applications. Springer, pp 111–120

  • Karp RM (1983) The probabilistic analysis of combinatorial optimization algorithms. In: Proceedings of the 10th international symposium on mathematical programming, pp 1601–1609

  • Keinänen, H (2009) Simulated annealing for multi-agent coalition formation. In: KES International symposium on agent and multi-agent systems: technologies and applications. Springer, pp 30–39

  • Larson KS, Sandholm TW (2000) Anytime coalition structure generation: an average case study. J Exp Theor Artif Intell 12(1):23–42

    Article  Google Scholar 

  • Michalak T, Rahwan T, Elkind E, Wooldridge M, Jennings NR (2016) A hybrid exact algorithm for complete set partitioning. Artif Intell 230:14–50

    Article  MathSciNet  Google Scholar 

  • Myerson RB (1977) Graphs and cooperation in games. Math Oper Res 2(3):225–229

    Article  MathSciNet  Google Scholar 

  • Norman TJ, Preece A, Chalmers S, Jennings NR, Luck M, Dang VD, Nguyen TD, Deora V, Shao J, Gray WA et al (2004) Agent-based formation of virtual organisations. Knowl-Based Syst 17(2):103–111

    Article  Google Scholar 

  • Rahwan T, Jennings NR (2008) An improved dynamic programming algorithm for coalition structure generation. In: Proceedings of the 7th international joint conference on Autonomous agents and multiagent systems, vol 3, AAMAS, pp 1417–1420

  • Rahwan T, Jennings NR (2008) Coalition structure generation: Dynamic programming meets anytime optimization. In: Proceedings of the 23rd AAAI conference on artificial intelligence, vol 8, AAAI, pp 156–161

  • Rahwan T, Ramchurn SD, Dang VD, Giovannucci A, Jennings NR (2007) Anytime optimal coalition structure generation. AAAI 7:1184–1190

    MATH  Google Scholar 

  • Rahwan T, Ramchurn SD, Jennings NR, Giovannucci A (2009) An anytime algorithm for optimal coalition structure generation. J Artif Intell Res 34:521–567

    Article  MathSciNet  Google Scholar 

  • Rahwan T, Michalak TP, Jennings NR (2012) A hybrid algorithm for coalition structure generation. In: Proceedings of the 26th AAAI conference on artificial intelligence, AAAI, pp 1443–1449

  • Rahwan T, Michalak TP, Wooldridge M, Jennings NR (2015) Coalition structure generation: a survey. Artif Intell 229:139–174

    Article  MathSciNet  Google Scholar 

  • Rothkopf MH, Pekeč A, Harstad RM (1998) Computationally manageable combinational auctions. Manag Sci 44(8):1131–1147

    Article  Google Scholar 

  • Sandhlom TW, Lesser VR (1997) Coalitions among computationally bounded agents. Artif Intell 94(1):99–137

    Article  MathSciNet  Google Scholar 

  • Sandholm T, Larson K, Andersson M, Shehory O, Tohmé F (1999) Coalition structure generation with worst case guarantees. Artif Intell 111(1):209–238

    Article  MathSciNet  Google Scholar 

  • Sen S, Dutta PS (2000) Searching for optimal coalition structures. In: Proceedings of the sixth international conference on multi-agent systems, ICMAS, pp 286–292

  • Service TC, Adams JA (2010) Anytime dynamic programming for coalition structure generation. In: Proceedings of the 9th international conference on autonomous agents and multiagent systems, vol 1, AAMAS, pp 1411–1412

  • Service TC, Adams JA (2011) Constant factor approximation algorithms for coalition structure generation. Auton Agent Multi-Agent Syst 23(1):1–17

    Article  Google Scholar 

  • Shehory O, Kraus S (1995) Coalition formation among autonomous agents: strategies and complexity (preliminary report). Springer, pp 55–72

  • Shehory O, Kraus S (1995) Task allocation via coalition formation among autonomous agents. In: IJCAI (1), pp 655–661

  • Shehory O, Kraus S (1998) Methods for task allocation via agent coalition formation. Artif Intell 101(1–2):165–200

    Article  MathSciNet  Google Scholar 

  • Vinyals M, Voice T, Ramchurn S, Jennings NR (2013) A hierarchical dynamic programming algorithm for optimal coalition structure generation. arXiv preprint arXiv:1310.6704

  • Voice T, Ramchurn SD, Jennings NR (2012) On coalition formation with sparse synergies. In: Proceedings of the 11th international conference on autonomous agents and multiagent systems, vol 1. International Foundation for Autonomous Agents and Multiagent Systems, pp 223–230

  • Yun Yeh D (1986) A dynamic programming approach to the complete set partitioning problem. BIT Numer Math 26(4):467–474

    Article  MathSciNet  Google Scholar 

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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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