Skip to main content

A Fine-Grained View on Stable Many-To-One Matching Problems with Lower and Upper Quotas

Part of the Lecture Notes in Computer Science book series (LNISA,volume 12495)


In the Hospital Residents problem with lower and upper quotas (HR- \({Q}_{L}^{U}\)), the goal is to find a stable matching of residents to hospitals where the number of residents matched to a hospital is either between its lower and upper quota or zero [Biró et al., TCS 2010]. We analyze this problem from a parameterized perspective using several natural parameters such as the number of hospitals and the number of residents. Moreover, we present a polynomial-time algorithm that finds a stable matching if it exists on instances with maximum lower quota two. Alongside HR- \({Q}_{L}^{U}\), we also consider two closely related models of independent interest, namely, the special case of HR- \({Q}_{L}^{U}\) where each hospital has only a lower quota but no upper quota and the variation of HR- \({Q}_{L}^{U}\) where hospitals do not have preferences over residents, which is also known as the House Allocation problem with lower and upper quotas.

We thank Robert Bredereck and Rolf Niedermeier for useful discussions.

N. Boehmer—supported by the DFG project MaMu (NI 369/19).

K. Heeger—supported by DFG Research Training Group 2434 “Facets of Complexity”.

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
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

Learn about institutional subscriptions


  1. Ágoston, K.C., Biró, P., McBride, I.: Integer programming methods for special college admissions problems. J. Comb. Optim. 32(4), 1371–1399 (2016).

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Arulselvan, A., Cseh, Á., Groß, M., Manlove, D.F., Matuschke, J.: Matchings with lower quotas: algorithms and complexity. Algorithmica 80(1), 185–208 (2018)

    CrossRef  MathSciNet  Google Scholar 

  3. Aziz, H., Gaspers, S., Sun, Z., Walsh, T.: From matching with diversity constraints to matching with regional quotas. In: AAMAS 2019, pp. 377–385 (2019)

    Google Scholar 

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

    Google Scholar 

  5. Biró, P., Fleiner, T., Irving, R.W., Manlove, D.: The college admissions problem with lower and common quotas. Theor. Comput. Sci. 411(34–36), 3136–3153 (2010)

    CrossRef  MathSciNet  Google Scholar 

  6. Biró, P., Manlove, D.F., McBride, I.: The hospitals/residents problem with couples: complexity and integer programming models. In: Gudmundsson, J., Katajainen, J. (eds.) SEA 2014. LNCS, vol. 8504, pp. 10–21. Springer, Cham (2014).

    CrossRef  Google Scholar 

  7. Boehmer, N., Heeger, K.: A fine-grained view on stable many-to-one matching problems with lower and upper quotas. CoRR abs/2009.14171 (2020)

    Google Scholar 

  8. Cechlárová, K., Fleiner, T.: Pareto optimal matchings with lower quotas. Math. Soc. Sci. 88, 3–10 (2017)

    CrossRef  MathSciNet  Google Scholar 

  9. Ehlers, L., Hafalir, I.E., Yenmez, M.B., Yildirim, M.A.: School choice with controlled choice constraints: hard bounds versus soft bounds. J. Econ. Theor. 153, 648–683 (2014)

    CrossRef  MathSciNet  Google Scholar 

  10. Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–15 (1962)

    CrossRef  MathSciNet  Google Scholar 

  11. Hamada, K., Iwama, K., Miyazaki, S.: The hospitals/residents problem with lower quotas. Algorithmica 74(1), 440–465 (2016)

    CrossRef  MathSciNet  Google Scholar 

  12. Irving, R.W.: An efficient algorithm for the “stable roommates” problem. J. Algorithms 6(4), 577–595 (1985)

    CrossRef  MathSciNet  Google Scholar 

  13. Kurata, R., Hamada, N., Iwasaki, A., Yokoo, M.: Controlled school choice with soft bounds and overlapping types. J. Artif. Intell. Res. 58, 153–184 (2017)

    CrossRef  MathSciNet  Google Scholar 

  14. Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)

    CrossRef  MathSciNet  Google Scholar 

  15. Manlove, D.F.: Algorithmics of Matching Under Preferences. Series on Theoretical Computer Science, vol. 2. World Scientific, Singapore (2013)

    CrossRef  Google Scholar 

  16. Marx, D., Schlotter, I.: Parameterized complexity and local search approaches for the stable marriage problem with ties. Algorithmica 58(1), 170–187 (2010)

    CrossRef  MathSciNet  Google Scholar 

  17. Marx, D., Schlotter, I.: Stable assignment with couples: parameterized complexity and local search. Discret. Optim. 8(1), 25–40 (2011)

    CrossRef  MathSciNet  Google Scholar 

  18. Mnich, M., Schlotter, I.: Stable matchings with covering constraints: a complete computational trichotomy. Algorithmica 82(5), 1136–1188 (2020)

    CrossRef  MathSciNet  Google Scholar 

  19. Monte, D., Tumennasan, N.: Matching with quorums. Econ. Lett. 120(1), 14–17 (2013)

    CrossRef  MathSciNet  Google Scholar 

  20. Ng, C., Hirschberg, D.S.: Three-dimensional stable matching problems. SIAM J. Discret. Math. 4(2), 245–252 (1991)

    CrossRef  MathSciNet  Google Scholar 

  21. Roth, A.E.: The evolution of the labor market for medical interns and residents: a case study in game theory. J. Polit. Econ. 92(6), 991–1016 (1984)

    CrossRef  Google Scholar 

  22. Roth, A.E.: Deferred acceptance algorithms: history, theory, practice, and open questions. Int. J. Game Theor. 36(3–4), 537–569 (2008)

    CrossRef  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Niclas Boehmer .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Boehmer, N., Heeger, K. (2020). A Fine-Grained View on Stable Many-To-One Matching Problems with Lower and Upper Quotas. In: Chen, X., Gravin, N., Hoefer, M., Mehta, R. (eds) Web and Internet Economics. WINE 2020. Lecture Notes in Computer Science(), vol 12495. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-64945-6

  • Online ISBN: 978-3-030-64946-3

  • eBook Packages: Computer ScienceComputer Science (R0)