Document Open Access Logo

Hedonic Games and Treewidth Revisited

Authors Tesshu Hanaka , Michael Lampis

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2022.64.pdf
  • Filesize: 0.76 MB
  • 16 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Hedonic Games
  • Nash Equilibrium
  • Treewidth

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We revisit the complexity of the well-studied notion of Additively Separable Hedonic Games (ASHGs). Such games model a basic clustering or coalition formation scenario in which selfish agents are represented by the vertices of an edge-weighted digraph G = (V,E), and the weight of an arc uv denotes the utility u gains by being in the same coalition as v. We focus on (arguably) the most basic stability question about such a game: given a graph, does a Nash stable solution exist and can we find it efficiently?
We study the (parameterized) complexity of ASHG stability when the underlying graph has treewidth t and maximum degree Δ. The current best FPT algorithm for this case was claimed by Peters [AAAI 2016], with time complexity roughly 2^{O(Δ⁵t)}. We present an algorithm with parameter dependence (Δ t)^{O(Δ t)}, significantly improving upon the parameter dependence on Δ given by Peters, albeit with a slightly worse dependence on t. Our main result is that this slight performance deterioration with respect to t is actually completely justified: we observe that the previously claimed algorithm is incorrect, and that in fact no algorithm can achieve dependence t^{o(t)} for bounded-degree graphs, unless the ETH fails. This, together with corresponding bounds we provide on the dependence on Δ and the joint parameter establishes that our algorithm is essentially optimal for both parameters, under the ETH.
We then revisit the parameterization by treewidth alone and resolve a question also posed by Peters by showing that Nash Stability remains strongly NP-hard on stars under additive preferences. Nevertheless, we also discover an island of mild tractability: we show that Connected Nash Stability is solvable in pseudo-polynomial time for constant t, though with an XP dependence on t which, as we establish, cannot be avoided.

Cite As Get BibTex

Tesshu Hanaka and Michael Lampis. Hedonic Games and Treewidth Revisited. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ESA.2022.64

Author Details

Tesshu Hanaka
  • Department of Informatics, Faculty of Information Science and Electrical Engineering, Kyushu University, Japan
Michael Lampis
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016, Paris, France

Funding

This work is partially supported by PRC CNRS JSPS project PARAGA (Parameterized Approximation Graph Algorithms) JPJSBP 120192912 and by JSPS KAKENHI Grant Number JP21K17707, JP21H05852, JP22H00513.

Acknowledgements

We want to thank Dr. Rémy Belmonte for helpful discussions.

References

  1. Alessandro Aloisio, Michele Flammini, and Cosimo Vinci. The impact of selfishness in hypergraph hedonic games. In The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020, pages 1766-1773. AAAI Press, 2020. URL: https://aaai.org/ojs/index.php/AAAI/article/view/5542.
  2. Haris Aziz, Florian Brandl, Felix Brandt, Paul Harrenstein, Martin Olsen, and Dominik Peters. Fractional hedonic games. ACM Trans. Economics and Comput., 7(2):6:1-6:29, 2019. URL: https://doi.org/10.1145/3327970.
  3. Haris Aziz, Felix Brandt, and Hans Georg Seedig. Computing desirable partitions in additively separable hedonic games. Artif. Intell., 195:316-334, 2013. Google Scholar
  4. Haris Aziz and Rahul Savani. Hedonic games. In Handbook of Computational Social Choice, pages 356-376. Cambridge University Press, 2016. Google Scholar
  5. Coralio Ballester. NP-completeness in hedonic games. Games Econ. Behav., 49(1):1-30, 2004. Google Scholar
  6. Nathanaël Barrot, Kazunori Ota, Yuko Sakurai, and Makoto Yokoo. Unknown agents in friends oriented hedonic games: Stability and complexity. In The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27 - February 1, 2019, pages 1756-1763. AAAI Press, 2019. URL: https://doi.org/10.1609/aaai.v33i01.33011756.
  7. Nathanaël Barrot and Makoto Yokoo. Stable and envy-free partitions in hedonic games. In Sarit Kraus, editor, Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI 2019, Macao, China, August 10-16, 2019, pages 67-73. ijcai.org, 2019. URL: https://doi.org/10.24963/ijcai.2019/10.
  8. Vittorio Bilò, Laurent Gourvès, and Jérôme Monnot. On a simple hedonic game with graph-restricted communication. In SAGT, volume 11801 of Lecture Notes in Computer Science, pages 252-265. Springer, 2019. Google Scholar
  9. Niclas Boehmer and Edith Elkind. Individual-based stability in hedonic diversity games. In The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020, pages 1822-1829. AAAI Press, 2020. URL: https://aaai.org/ojs/index.php/AAAI/article/view/5549.
  10. Felix Brandt, Martin Bullinger, and Anaëlle Wilczynski. Reaching individually stable coalition structures in hedonic games. In Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty-Third Conference on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event, February 2-9, 2021, pages 5211-5218. AAAI Press, 2021. URL: https://ojs.aaai.org/index.php/AAAI/article/view/16658.
  11. Simina Brânzei and Kate Larson. Coalitional affinity games and the stability gap. In IJCAI, pages 79-84, 2009. Google Scholar
  12. Martin Bullinger and Stefan Kober. Loyalty in cardinal hedonic games. In Zhi-Hua Zhou, editor, Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI 2021, Virtual Event / Montreal, Canada, 19-27 August 2021, pages 66-72. ijcai.org, 2021. URL: https://doi.org/10.24963/ijcai.2021/10.
  13. Katarína Cechlárová. Stable partition problem. In Encyclopedia of Algorithms, pages 2075-2078. Springer, 2016. Google Scholar
  14. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  15. Andreas Darmann, Edith Elkind, Sascha Kurz, Jérôme Lang, Joachim Schauer, and Gerhard J. Woeginger. Group activity selection problem with approval preferences. Int. J. Game Theory, 47(3):767-796, 2018. URL: https://doi.org/10.1007/s00182-017-0596-4.
  16. Vladimir G. Deineko and Gerhard J. Woeginger. Two hardness results for core stability in hedonic coalition formation games. Discret. Appl. Math., 161(13-14):1837-1842, 2013. Google Scholar
  17. Edith Elkind, Angelo Fanelli, and Michele Flammini. Price of pareto optimality in hedonic games. Artif. Intell., 288:103357, 2020. Google Scholar
  18. Edith Elkind and Michael J. Wooldridge. Hedonic coalition nets. In AAMAS (1), pages 417-424. IFAAMAS, 2009. Google Scholar
  19. Angelo Fanelli, Gianpiero Monaco, and Luca Moscardelli. Relaxed core stability in fractional hedonic games. In Zhi-Hua Zhou, editor, Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI 2021, Virtual Event / Montreal, Canada, 19-27 August 2021, pages 182-188. ijcai.org, 2021. URL: https://doi.org/10.24963/ijcai.2021/26.
  20. Michele Flammini, Bojana Kodric, Gianpiero Monaco, and Qiang Zhang. Strategyproof mechanisms for additively separable and fractional hedonic games. J. Artif. Intell. Res., 70:1253-1279, 2021. Google Scholar
  21. Martin Gairing and Rahul Savani. Computing stable outcomes in symmetric additively separable hedonic games. Math. Oper. Res., 44(3):1101-1121, 2019. Google Scholar
  22. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  23. Tesshu Hanaka, Hironori Kiya, Yasuhide Maei, and Hirotaka Ono. Computational complexity of hedonic games on sparse graphs. In PRIMA, volume 11873 of Lecture Notes in Computer Science, pages 576-584. Springer, 2019. Google Scholar
  24. Ararat Harutyunyan, Michael Lampis, and Nikolaos Melissinos. Digraph coloring and distance to acyclicity. In STACS, volume 187 of LIPIcs, pages 41:1-41:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  25. Samuel Ieong and Yoav Shoham. Marginal contribution nets: a compact representation scheme for coalitional games. In EC, pages 193-202. ACM, 2005. Google Scholar
  26. Ayumi Igarashi and Edith Elkind. Hedonic games with graph-restricted communication. In AAMAS, pages 242-250. ACM, 2016. Google Scholar
  27. Ayumi Igarashi, Kazunori Ota, Yuko Sakurai, and Makoto Yokoo. Robustness against agent failure in hedonic games. In Sarit Kraus, editor, Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI 2019, Macao, China, August 10-16, 2019, pages 364-370. ijcai.org, 2019. URL: https://doi.org/10.24963/ijcai.2019/52.
  28. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  29. Michael Lampis. Minimum stable cut and treewidth. In ICALP, volume 198 of LIPIcs, pages 92:1-92:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  30. Michael Lampis, Stefan Mengel, and Valia Mitsou. QBF as an alternative to Courcelle’s theorem. In Olaf Beyersdorff and Christoph M. Wintersteiger, editors, Theory and Applications of Satisfiability Testing - SAT 2018 - 21st International Conference, SAT 2018, Held as Part of the Federated Logic Conference, FloC 2018, Oxford, UK, July 9-12, 2018, Proceedings, volume 10929 of Lecture Notes in Computer Science, pages 235-252. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-94144-8_15.
  31. Michael Lampis and Valia Mitsou. Treewidth with a quantifier alternation revisited. In IPEC, volume 89 of LIPIcs, pages 26:1-26:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  32. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. SIAM J. Comput., 47(3):675-702, 2018. Google Scholar
  33. Kazunori Ohta, Nathanaël Barrot, Anisse Ismaili, Yuko Sakurai, and Makoto Yokoo. Core stability in hedonic games among friends and enemies: Impact of neutrals. In Carles Sierra, editor, Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017, Melbourne, Australia, August 19-25, 2017, pages 359-365. ijcai.org, 2017. URL: https://doi.org/10.24963/ijcai.2017/51.
  34. Martin Olsen. Nash stability in additively separable hedonic games and community structures. Theory Comput. Syst., 45(4):917-925, 2009. Google Scholar
  35. Martin Olsen, Lars Bækgaard, and Torben Tambo. On non-trivial nash stable partitions in additive hedonic games with symmetric 0/1-utilities. Inf. Process. Lett., 112(23):903-907, 2012. Google Scholar
  36. Dominik Peters. Graphical hedonic games of bounded treewidth. In AAAI, pages 586-593. AAAI Press, 2016. Google Scholar
  37. Dominik Peters. Precise complexity of the core in dichotomous and additive hedonic games. In ADT, volume 10576 of Lecture Notes in Computer Science, pages 214-227. Springer, 2017. Google Scholar
  38. Dominik Peters and Edith Elkind. Simple causes of complexity in hedonic games. In IJCAI, pages 617-623. AAAI Press, 2015. Google Scholar
  39. Walid Saad, Zhu Han, Tamer Basar, Mérouane Debbah, and Are Hjørungnes. Hedonic coalition formation for distributed task allocation among wireless agents. IEEE Trans. Mob. Comput., 10(9):1327-1344, 2011. URL: https://doi.org/10.1109/TMC.2010.242.
  40. Jakub Sliwinski and Yair Zick. Learning hedonic games. In Carles Sierra, editor, Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017, Melbourne, Australia, August 19-25, 2017, pages 2730-2736. ijcai.org, 2017. URL: https://doi.org/10.24963/ijcai.2017/380.
  41. Shao Chin Sung and Dinko Dimitrov. Computational complexity in additive hedonic games. Eur. J. Oper. Res., 203(3):635-639, 2010. Google Scholar
  42. Gerhard J. Woeginger. A hardness result for core stability in additive hedonic games. Math. Soc. Sci., 65(2):101-104, 2013. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact