SpringerLink
Log in

Stable Matching with Multilayer Approval Preferences: Approvals Can Be Harder Than Strict Preferences

  • Conference paper
  • First Online:
Algorithmic Game Theory (SAGT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13584))

Included in the following conference series:

  • 726 Accesses

Abstract

We study stable matching problems where agents have multilayer preferences: There are \(\ell \) layers each consisting of one preference relation for each agent. Recently, Chen et al. [EC ’18] studied such problems with strict preferences, establishing four multilayer adaptations of classical notions of stability. We follow up on their work by analyzing the computational complexity of stable matching problems with multilayer approval preferences. We consider eleven stability notions derived from three well-established stability notions for stable matchings with ties and the four adaptations proposed by Chen et al. For each stability notion, we show that the problem of finding a stable matching is either polynomial-time solvable or NP-hard. Furthermore, we examine the influence of the number of layers and the desired “degree of stability” on the problems’ complexity. Somewhat surprisingly, we discover that assuming approval preferences instead of strict preferences does not considerably simplify the situation (and sometimes even makes polynomial-time solvable problems NP-hard).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (France)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 42.79
Price includes VAT (France)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
EUR 52.74
Price includes VAT (France)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    In our model, agents prefer agents they approve to agents they disapprove and to having no partner, but are indifferent between the later two.

  2. 2.

    E.g., a weakly stable matching corresponds to a maximal matching in the undirected part of the approval graph (see Sect. 2).

  3. 3.

    For individual stability, we have only two stability notions depending on the definition of when an agent “favors” another agent.

  4. 4.

    These are finding an \(\ell \)-individually stable matching (Theorem 2) and finding an \(\ell \)-pair stable matching for bipartite approvals where the preferences of agents on one side do not change (Theorem 1).

  5. 5.

    We call it individual “weak/super” stability since it coincides with weak/super stability for \(\ell =1\).

  6. 6.

    For example, let \(A=\{a_1,a_2,a_3,a_4\}\) and \(\ell =2\) with \(a_1\) and \(a_2\) approving each other in the first layer and \(a_3\) and \(a_4\) approving each other in the second layer. Then, \(M=\{\{a_1,a_2\},\{a_3,a_4\}\}\) is all-layers super stable but not \(\ell \)-individually super stable. Modifying the instance by letting \(a_1\) and \(a_3\) approve each other in both layers, M is all-layers weakly stable but not \(\ell \)-individually weakly stable.

References

  1. Aziz, H., et al.: Stable matching with uncertain pairwise preferences. In: Proceedings of AAMAS-2017, pp. 344–352. ACM (2017)

    Google Scholar 

  2. Aziz, H., Biró, P., Gaspers, S., de Haan, R., Mattei, N., Rastegari, B.: Stable matching with uncertain linear preferences. Algorithmica 82(5), 1410–1433 (2020)

    Article  MathSciNet  Google Scholar 

  3. Aziz, H., Biró, P., de Haan, R., Rastegari, B.: Pareto optimal allocation under uncertain preferences: uncertainty models, algorithms, and complexity. Artif. Intell. 276, 57–78 (2019)

    Article  MathSciNet  Google Scholar 

  4. Aziz, H., Bogomolnaia, A., Moulin, H.: Fair mixing: the case of dichotomous preferences. ACM Trans. Econ. Comput. 8(4), 18:1–18:27 (2020)

    Google Scholar 

  5. Aziz, H., Savani, R.: Hedonic games. In: Handbook of Computational Social Choice, pp. 356–376. Cambridge University Press, Cambridge (2016)

    Google Scholar 

  6. Bentert, M., Boehmer, N., Heeger, K., Koana, T.: Stable matching with multilayer approval preferences: approvals can be harder than strict preferences. CoRR abs/2205.07550 (2022)

    Google Scholar 

  7. Boccaletti, S., et al.: The structure and dynamics of multilayer networks. Phys. Rep. 544(1), 1–122 (2014)

    Article  MathSciNet  Google Scholar 

  8. Boehmer, N., Brill, M., Schmidt-Kraepelin, U.: Proportional representation in matching markets: selecting multiple matchings under dichotomous preferences. In: Proceedings of AAMAS-2022, pp. 136–144. IFAAMAS (2022)

    Google Scholar 

  9. Boehmer, N., Heeger, K., Niedermeier, R.: Deepening the (parameterized) complexity analysis of incremental stable matching problems. In: Proceedings of MFCS-2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022, accepted for publication)

    Google Scholar 

  10. Boehmer, N., Heeger, K., Niedermeier, R.: Theory of and experiments on minimally invasive stability preservation in changing two-sided matching markets. In: Proceedings of AAAI-2022, pp. 4851–4858. AAAI Press (2022)

    Google Scholar 

  11. Boehmer, N., Niedermeier, R.: Broadening the research agenda for computational social choice: multiple preference profiles and multiple solutions. In: Proceedings of AAMAS-2021, pp. 1–5. ACM (2021)

    Google Scholar 

  12. Bouveret, S., Lang, J.: Efficiency and envy-freeness in fair division of indivisible goods: Logical representation and complexity. J. Artif. Intell. Res. 32, 525–564 (2008)

    Article  MathSciNet  Google Scholar 

  13. Bredereck, R., Heeger, K., Knop, D., Niedermeier, R.: Multidimensional stable roommates with master list. In: Chen, X., Gravin, N., Hoefer, M., Mehta, R. (eds.) WINE 2020. LNCS, vol. 12495, pp. 59–73. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64946-3_5

    Chapter  Google Scholar 

  14. Bredereck, R., Komusiewicz, C., Kratsch, S., Molter, H., Niedermeier, R., Sorge, M.: Assessing the computational complexity of multilayer subgraph detection. Netw. Sci. 7(2), 215–241 (2019)

    Article  Google Scholar 

  15. Chen, J., Niedermeier, R., Skowron, P.: Stable marriage with multi-modal preferences. In: Proceedings of EC-2018, pp. 269–286. ACM (2018)

    Google Scholar 

  16. Irving, R.W.: Stable marriage and indifference. Discret. Appl. Math. 48(3), 261–272 (1994)

    Article  MathSciNet  Google Scholar 

  17. Irving, R.W., Manlove, D.F.: The stable roommates problem with ties. J. Algorithms 43(1), 85–105 (2002)

    Article  MathSciNet  Google Scholar 

  18. Irving, R.W., Manlove, D.F., Scott, S.: The stable marriage problem with master preference lists. Discret. Appl. Math. 156(15), 2959–2977 (2008)

    Article  MathSciNet  Google Scholar 

  19. Jain, P., Talmon, N.: Committee selection with multimodal preferences. In: Proceedings of ECAI-2020, pp. 123–130. IOS Press (2020)

    Google Scholar 

  20. Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. J. Complex Netw. 2(3), 203–271 (2014)

    Article  Google Scholar 

  21. Kunysz, A.: The strongly stable roommates problem. In: Proceedings of ESA-2016, pp. 60:1–60:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

    Google Scholar 

  22. Kyropoulou, M., Suksompong, W., Voudouris, A.A.: Almost envy-freeness in group resource allocation. In: Proceedings of IJCAI-2019, pp. 400–406. ijcai.org (2019)

    Google Scholar 

  23. Lackner, M., Skowron, P.: Approval-based committee voting: axioms, algorithms, and applications. CoRR abs/2007.01795 (2020)

    Google Scholar 

  24. Magnani, M., Rossi, L.: The ML-model for multi-layer social networks. In: Proceedings of ASONAM-2011, pp. 5–12. IEEE Computer Society (2011)

    Google Scholar 

  25. Manlove, D.F.: Algorithmics of Matching Under Preferences, Series on Theoretical Computer Science, vol. 2. WorldScientific (2013)

    Google Scholar 

  26. Meeks, K., Rastegari, B.: Solving hard stable matching problems involving groups of similar agents. Theor. Comput. Sci. 844, 171–194 (2020)

    Article  MathSciNet  Google Scholar 

  27. Miyazaki, S., Okamoto, K.: Jointly stable matchings. J. Comb. Optim. 38(2), 646–665 (2019). https://doi.org/10.1007/s10878-019-00402-4

    Article  MathSciNet  MATH  Google Scholar 

  28. Peters, D.: Complexity of hedonic games with dichotomous preferences. In: Proceedings of AAAI-2016, pp. 579–585. AAAI Press (2016)

    Google Scholar 

  29. Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of STOC-1978, pp. 216–226. ACM (1978)

    Google Scholar 

  30. Segal-Halevi, E., Suksompong, W.: Democratic fair allocation of indivisible goods. Artif. Intell. 277, 103–167 (2019)

    Article  MathSciNet  Google Scholar 

  31. Steindl, B., Zehavi, M.: Parameterized analysis of assignment under multiple preferences. In: Rosenfeld, A., Talmon, N. (eds.) EUMAS 2021. LNCS (LNAI), vol. 12802, pp. 160–177. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-82254-5_10

    Chapter  Google Scholar 

  32. Steindl, B., Zehavi, M.: Verification of multi-layered assignment problems. In: Rosenfeld, A., Talmon, N. (eds.) EUMAS 2021. LNCS (LNAI), vol. 12802, pp. 194–210. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-82254-5_12

    Chapter  Google Scholar 

  33. Suksompong, W.: Approximate maximin shares for groups of agents. Math. Soc. Sci. 92, 40–47 (2018)

    Article  MathSciNet  Google Scholar 

  34. Talmon, N., Faliszewski, P.: A framework for approval-based budgeting methods. In: Proceedings of AAAI-2019, pp. 2181–2188. AAAI Press (2019)

    Google Scholar 

  35. Wen, Y., Zhou, A., Guo, J.: Position-based matching with multi-modal preferences. In: Proceedings of AAMAS-2022, pp. 1373–1381. IFAAMAS (2022)

    Google Scholar 

Download references

Acknowledgements

MB was supported by the DFG project MaMu (NI 369/19). NB was supported by the DFG project MaMu (NI 369/19) and by the DFG project ComSoc-MPMS (NI 369/22). KH was supported by the DFG Research Training Group 2434 “Facets of Complexity” and by the DFG project FPTinP (NI 369/16). TK was supported by the DFG project DiPa (NI 369/21). This work was started at the research retreat of the TU Berlin Algorithmics and Computational Complexity research group held in Zinnowitz (Usedom) in September 2021.

Author information

Authors and Affiliations

  1. Algorithmics and Computational Complexity, Technische Universität Berlin, Berlin, Germany

    Matthias Bentert, Niclas Boehmer, Klaus Heeger & Tomohiro Koana

Authors
  1. Matthias Bentert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Niclas Boehmer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Klaus Heeger
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tomohiro Koana
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Niclas Boehmer .

Editor information

Editors and Affiliations

  1. University of Essex, Colchester, UK

    Panagiotis Kanellopoulos

  2. University of Essex, Colchester, UK

    Maria Kyropoulou

  3. University of Essex, Colchester, UK

    Alexandros Voudouris

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bentert, M., Boehmer, N., Heeger, K., Koana, T. (2022). Stable Matching with Multilayer Approval Preferences: Approvals Can Be Harder Than Strict Preferences. In: Kanellopoulos, P., Kyropoulou, M., Voudouris, A. (eds) Algorithmic Game Theory. SAGT 2022. Lecture Notes in Computer Science, vol 13584. Springer, Cham. https://doi.org/10.1007/978-3-031-15714-1_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-15714-1_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-15713-4

  • Online ISBN: 978-3-031-15714-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation