Skip to main content
SpringerLink
Log in

Maximum Stable Matching with One-Sided Ties of Bounded Length

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We study the problem of finding maximum weakly stable matchings when preference lists are incomplete and contain one-sided ties of bounded length. We show that if the tie length is at most L, then it is possible to achieve an approximation ratio of \(1 + (1 - \frac {1}{L})^{L}\). We also show that the same ratio is an upper bound on the integrality gap, which matches the known lower bound. In the case where the tie length is at most 2, our result implies an approximation ratio and integrality gap of \(\frac {5}{4}\), which matches the known UG-hardness result.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In the proof of this claim [22, Theorem 19], Huang et al. exhibit a family of instances with 2k men and 2k women such that the corresponding LP has a feasible fractional value of (3/2 − o(1))k. It is asserted that a certain weakly stable matching of size k is a maximum weakly stable matching, but this assertion is incorrect. For the case when k = 2, there exists a weakly stable matching of size 3. Similarly, when k > 2, it can be shown that the maximum size of weakly stable matching is greater than k.

    There is also a flaw related to the main result of their paper, which asserts an approximation ratio of \(\frac {5}{4}\) for the special case where ties are one-sided and are restricted to the end of the preference lists. In the derivation of inequalities (11) and (12) in their proof [22, Lemma 16], it is claimed that \(\frac {\delta _{m, w}}{1 + \nu _{w}} \leq \delta _{m, w}\). This claim depends on the unproven assumption that δm, w is non-negative. It is unclear whether this flaw can be fixed. Both flaws have been acknowledged by Huang et al. in a personal communication.

  2. A binary relation ≽ over a set K is said to satisfy antisymmetry if for every k1, k2 ∈ K such that k1 ≽ k2 and k2 ≽ k1, we have k1 = k2.

  3. A binary relation ≽ over a set K is said to satisfy transitivity if for every k1, k2, k3 ∈ K such that k1 ≽ k2 and k2 ≽ k3, we have k1 ≽ k3.

  4. A binary relation ≽ over a set K is said to satisfy totality if for every k1, k2 ∈ K, we have either k1 ≽ k2 or k2 ≽ k1.

  5. The asymmetric part of a binary relation ≽ over a set K is the binary relation ≻ over the set K such that for every k1, k2 ∈ K, we have k1 ≻ k2 if and only if k1 ≽ k2 and \(\lnot (k_{2} \succeq k_{1})\).

  6. The symmetric part of a binary relation ≽ over a set K is the binary relation \(\sim \) over the set K such that for every k1, k2 ∈ K, we have \(k_{1} \sim k_{2}\) if and only if k1 ≽ k2 and k2 ≽ k1.

  7. Some of the literature on stable matching with indifferences does not allow an agent to be indifferent between being matched to an agent and being unmatched. Our formulation of the stable matching problem with one-sided ties and incomplete lists allows for this possibility, since we can have i =j0 for any man i and woman j.

  8. For any positive step size η, the upper bound on pi can be strengthened to pi < 1 + 2η. Later in the paper, we present an algorithm (Algorithm 1) that terminates in a state satisfying (P1) through (P4) with η = 0 (Lemma 13); in that context, we have pi ≤ 1 and not pi < 1.

References

  1. Lam, C-K, Plaxton, C.G.: A (1 + 1/e)-approximation algorithm for maximum stable matching with one-sided ties and incomplete lists. In: Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms, pp 2823–2840 (2019)

  2. Lam, C-K, Plaxton, C.G.: Maximum stable matching with one-sided ties of bounded length. In: Proceedings of the 12th International Symposium on Algorithmic Game Theory, pp 343–356 (2019)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Gale, D., Sotomayor, M.A.O.: Some remarks on the stable matching problem. Discret. Appl. Math. 11(3), 223–232 (1985)

    Article  MathSciNet  Google Scholar 

  6. 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)

    Article  Google Scholar 

  7. Manlove, D.F., Irving, R.W., Iwama, K., Miyazaki, S., Morita, Y.: Hard variants of stable marriage. Theor. Comput. Sci. 276(1), 261–279 (2002)

    Article  MathSciNet  Google Scholar 

  8. Iwama, K., Miyazaki, S., Yamauchi, N.: A 1.875-approximation algorithm for the stable marriage problem. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp 288–297 (2007)

  9. Király, Z.: Better and simpler approximation algorithms for the stable marriage problem. Algorithmica 60(1), 3–20 (2011)

    Article  MathSciNet  Google Scholar 

  10. McDermid, E.J.: A 3/2-approximation algorithm for general stable marriage. In: Proceedings of the 36th International Colloquium on Automata, Languages and Programming, pp 689–700 (2009)

  11. Paluch, K.: Faster and simpler approximation of stable matchings. Algorithms 7(2), 189–202 (2014)

    Article  MathSciNet  Google Scholar 

  12. Király, Z.: Linear time local approximation algorithm for maximum stable marriage. Algorithms 6(3), 471–484 (2013)

    Article  MathSciNet  Google Scholar 

  13. Iwama, K., Manlove, D.F., Miyazaki, S., Morita, Y.: Stable marriage with incomplete lists and ties. In: Proceedings of the 26th International Colloquium on Automata, Languages, and Programming, pp 443–452 (1999)

  14. Halldórsson, M.M., Irving, R.W., Iwama, K., Manlove, D.F., Miyazaki, S., Morita, Y., Scott, S.: Approximability results for stable marriage problems with ties. Theor. Comput. Sci. 306(1), 431–447 (2003)

    Article  MathSciNet  Google Scholar 

  15. Yanagisawa, H.: Approximation algorithms for stable marriage problems. Ph.D. Thesis, Graduate School of Informatics, Kyoto University (2007)

  16. Iwama, K., Miyazaki, S., Yanagisawa, H.: A 25/17-approximation algorithm for the stable marriage problem with one-sided ties. Algorithmica 68 (3), 758–775 (2014)

    Article  MathSciNet  Google Scholar 

  17. Dean, B.C., Jalasutram, R.: Factor revealing LPs and stable matching with ties and incomplete lists. In: Proceedings of the 3rd International Workshop on Matching Under Preferences, pp 42–53 (2015)

  18. Huang, C-C, Kavitha, T.: Improved approximation algorithms for two variants of the stable marriage problem with ties. Math. Program. 154(1), 353–380 (2015)

    Article  MathSciNet  Google Scholar 

  19. Bauckholt, F., Pashkovich, K., Sanità, L.: On the approximability of the stable marriage problem with one-sided ties. arXiv:1805.05391(2018)

  20. Radnai, A.: Approximation algorithms for the stable marriage problem. Master’s Thesis, Department of Computer Science, Eötvös Loránd University (2014)

  21. Halldórsson, M.M., Iwama, K., Miyazaki, S., Yanagisawa, H.: Improved approximation results for the stable marriage problem. ACM Trans. Algorithm. 3(3), 30 (2007)

    Article  MathSciNet  Google Scholar 

  22. Huang, C-C, Iwama, K., Miyazaki, S., Yanagisawa, H.: A tight approximation bound for the stable marriage problem with restricted ties. In: Proceedings of the 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pp 361–380 (2015)

  23. Halldórsson, M.M., Iwama, K., Miyazaki, S., Yanagisawa, H.: Randomized approximation of the stable marriage problem. Theor. Comput. Sci. 325 (3), 439–465 (2004)

    Article  Google Scholar 

  24. Chiang, R., Pashkovich, K.: On the approximability of the stable matching problem with ties of size two. arXiv:1808.04510 (2019)

  25. Irving, R.W., Manlove, D.F., O’Malley, G.: Stable marriage with ties and bounded length preference lists. J. Discret. Algorithm. 7(2), 213–219 (2009)

    Article  MathSciNet  Google Scholar 

  26. Abraham, D.J., Levavi, A., Manlove, D.F., O’Malley, G.: The stable roommates problem with globally-ranked pairs. In: Proceedings of the 3rd International Workshop on Web and Internet Economics, pp 431–444 (2007)

  27. 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 

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

    Article  MathSciNet  Google Scholar 

  29. Hamada, K., Miyazaki, S., Yanagisawa, H.: Strategy-proof approximation algorithms for the stable marriage problem with ties and incomplete lists. In: Proceedings of the 30th International Symposium on Algorithms and Computation, pp 9:1–9:14 (2019)

  30. Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. ACM 50(6), 795–824 (2003)

    Article  MathSciNet  Google Scholar 

  31. Fernandes, C.G., Meira, L.A.A., Miyazawa, F.K., Pedrosa, L.L.C.: A systematic approach to bound factor-revealing LPs and its application to the metric and squared metric facility location problems. Math. Program. 153(2), 655–685 (2015)

    Article  MathSciNet  Google Scholar 

  32. Mahdian, M., Yan, Q.: Online bipartite matching with random arrivals: An approach based on strongly factor-revealing LPs. In: Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, pp 597–606 (2011)

  33. Mehta, A., Saberi, A., Vazirani, U., Vazirani, V.: Adwords and generalized online matching. J. ACM 54(5), 22:1–22:19 (2007)

    Article  MathSciNet  Google Scholar 

  34. Rothblum, U.G.: Characterization of stable matchings as extreme points of a polytope. Math. Program. 54(1), 57–67 (1992)

    Article  MathSciNet  Google Scholar 

  35. Vande Vate, J.H.: Linear programming brings marital bliss. Oper. Res. Lett. 8(3), 147–153 (1989)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We thank the two anonymous referees for their valuable comments, which helped us to significantly improve the presentation.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Texas at Austin, Austin, TX, USA

    Chi-Kit Lam & C. Gregory Plaxton

Authors
  1. Chi-Kit Lam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Gregory Plaxton
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Gregory Plaxton.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article belongs to the Topical Collection: Special Issue on Algorithmic Game Theory (SAGT 2019)

Guest Editors: Dimitris Fotakis and Vangelis Markakis

This journal article combines the results presented in two conference papers [1, 2]. The first of these papers establishes an approximation ratio of \(1+\protect \frac {1}{e}\). The second establishes the more detailed bound stated in the abstract. The latter bound is increasing in L and approaches \(1+\protect \frac {1}{e}\) as L tends to infinity.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lam, CK., Plaxton, C.G. Maximum Stable Matching with One-Sided Ties of Bounded Length. Theory Comput Syst 66, 645–678 (2022). https://doi.org/10.1007/s00224-022-10072-1

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-022-10072-1

Keywords

Search

Navigation