Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Popular Solutions for Optimal Matchings

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14760))

Included in the following conference series:

  • 132 Accesses

Abstract

Let G be a bipartite graph where every vertex has a strict preference order over its neighbors. The preferences of a vertex over its neighbors extend naturally to preferences over matchings. A matching M is popular in G if there is no matching N such that vertices that prefer N outnumber those that prefer M. Every stable matching is popular. We consider the following variant: edges in G have utilities and it is only max-utility matchings that are relevant for us. We show there always exists a max-utility matching that is popular within the set of all max-utility matchings; moreover, such a matching can be efficiently computed. We focus on largest max-utility matchings and show a compact extended formulation for the polytope of largest max-utility matchings that are popular within the set of all largest max-utility matchings.

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
$34.99 /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
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 64.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    If \(\textsf{wt}_M(e) \ne -2\), then \(\textsf{wt}_M(e) \ge 0\). Since \(e \notin M\), this means that at least one of a, b prefers the other to its partner in M. If a prefers b to its partner in M then \(e_i\) blocks \(M'\); if b prefers a to its assignment in M then \(e_{i-1}\) blocks \(M'\).

References

  1. Ahani, N., Andersson, T., Martinello, A., Trapp, A., Teytelboym, A.: Placement optimization in refugee resettlement. Oper. Res. 69(5), 1468–1486 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  2. Biro, P., Manlove, D.F., Mittal, S.: Size versus stability in the marriage problem. Theoret. Comput. Sci. 411, 1828–1841 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cseh, A.: Popular matchings. In: Endriss, U. (ed.) Trends in Computational Social Choice (2017)

    Google Scholar 

  4. Dulmage, A.L., Mendelsohn, N.S.: Coverings of bipartite graphs. Can. J. Math. 10, 517–534 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  5. Faenza, Y., Kavitha, T.: Quasi-popular matchings, optimality, and extended formulations. Math. Oper. Res. 47(1), 427–457 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  6. Faenza, Y., Kavitha, T., Powers, V., Zhang, X.: Popular matchings and limits to tractability. In: Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 2790–2809 (2019)

    Google Scholar 

  7. Gale, D., Shapley, L.: College admissions and the stability of marriage. Amer. Math. Monthly 69(1), 9–15 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gale, D., Sotomayor, M.: Some remarks on the stable matching problem. Discret. Appl. Math. 11, 223–232 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behav. Sci. 20, 166–173 (1975)

    Article  MATH  Google Scholar 

  10. Goemans, M.: Combinatorial optimization (2017). https://math.mit.edu/~goemans/18453S17/18453.html

  11. Kavitha, T.: A size-popularity tradeoff in the stable marriage problem. SIAM J. Comput. 43(1), 52–71 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kavitha, T.: Matchings, critical nodes, and popular solutions. In: Proceedings of the 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS, pp. 25:1–25:19 (2021)

    Google Scholar 

  13. Kavitha, T.: Maximum matchings and popularity. SIAM J. Discr. Math. 38(2), 1202–1221 (2024). https://doi.org/10.1137/22M1523248

    Article  MathSciNet  MATH  Google Scholar 

  14. Losász, L., Plummer, M.D.: Matching Theory. Mathematics Studies, vol. 121. North-Holland (1986)

    Google Scholar 

  15. Nasre, M., Nimbhorkar, P., Ranjan, K., Sarkar, A.: Popular matchings in the hospital-residents problem with two-sided lower quotas. In: Proceedings of the 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS, pp. 30:1–30:21 (2021)

    Google Scholar 

  16. Pulleyblank, W.R.: Chapter 3, matchings and extensions. In: Graham, R.L., Grötschel, M., Lovasz, L. (ed.) The Handbook of Combinatorics (1995)

    Google Scholar 

  17. Robards, P.A.: Applying the two-sided matching processes to the United States Navy enlisted assignment process. Master’s thesis, Naval Postgraduate School, Monterey, Canada (2001)

    Google Scholar 

  18. Rothblum, U.G.: Characterization of stable matchings as extreme points of a polytope. Math. Program. 54, 57–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  19. Soldner, M.: Optimization and measurement in humanitarian operations: addressing practical needs. Ph.D. thesis, Georgia Institute of Technology (2014)

    Google Scholar 

  20. Teo, C.P., Sethuraman, J.: The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4), 874–891 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yang, W., Giampapa, J.A., Sycara, K.: Two-sided matching for the US Navy detailing process with market complication. Technical report CMU-R1-TR-03-49, Robotics Institute, Carnegie Mellon University (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tata Institute of Fundamental Research, Mumbai, India

    Telikepalli Kavitha

Authors
  1. Telikepalli Kavitha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Telikepalli Kavitha .

Editor information

Editors and Affiliations

  1. Masaryk University, Brno, Czech Republic

    Daniel Kráľ

  2. University of Primorska, Koper, Slovenia

    Martin Milanič

Rights and permissions

Reprints and permissions

Copyright information

© 2025 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

Kavitha, T. (2025). Popular Solutions for Optimal Matchings. In: Kráľ, D., Milanič, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2024. Lecture Notes in Computer Science, vol 14760. Springer, Cham. https://doi.org/10.1007/978-3-031-75409-8_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-75409-8_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-75408-1

  • Online ISBN: 978-3-031-75409-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation