Skip to main content
SpringerLink
Log in
Book cover

Mathematical Programming and Game Theory pp 141–157Cite as

Some Hard Stable Marriage Problems: A Survey on Multivariate Analysis

  • Chapter
  • First Online:

  • 1188 Accesses

Part of the book series: Indian Statistical Institute Series ((INSIS))

Abstract

We survey an emerging area of research within algorithmic game theory: multivariate analysis of games. This article surveys the landscape of work on various stable marriage problems and the use of parametrized complexity as a toolbox to study computationally hard variants of these problems. Our survey can be divided into three broad topics: strategic manipulation, maximum (minimum) sized matching in the presence of ties, and notions of fair or equitable stable matchings.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Notes

  1. 1.

    If a parameterized problem cannot be solved in polynomial time even when the value of the parameter is a fixed constant (that is, independent of the input), then the problem is said to be paraNP-hard.

  2. 2.

    In the analysis of the Balanced Stable Marriage, it is assumed that any stable matching is perfect.

  3. 3.

    In Stable Roommate, the matching market consists of agents of the same type, as opposed to the market modeled the stable marriage problem that consists of agents of two types, men and women. Roommate assignments in college housing facilities is a real-world application of the stable roommate problem.

  4. 4.

    Two instances \(\mathscr {I}\) and \(\mathscr {J}\) are said to be equivalent if \(\mathscr {I}\) is a Yes-instance if and only if \(\mathscr {J}\) is a Yes-instance.

References

  1. Dan Gusfield, D., Irving, R.W.: The Stable Marriage Problem-Structure and Algorithm. MIT Press, Cambridge (1989)

    Google Scholar 

  2. Knuth, D. E.: Stable marriage and its relation to other combinatorial problems: an introduction to the mathematical analysis of algorithms. In: CRM Proceedings & Lecture Notes. American Mathematical Society, Providence, R.I. (1997)

    Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Myerson, R.B.: Graphs and cooperation games. Math. Op. Res. 2, 225–229 (1977)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Manlove, David F., D.F.: The structure of stable marriage with indifference. Discret. Appl. Math. 122, 167–181 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Irving, R.W., Leather, P., Gusfield, D.: An efficient algorithm for the “optimal” stable marriage. J. ACM 34, 532–543 (1987)

    Article  MathSciNet  Google Scholar 

  10. Kato, A.: Complexity of the sex-equal stable marriage problem. Jpn. J. Ind. Appl. Math. 10, 1 (1993)

    Article  MathSciNet  Google Scholar 

  11. McDermid, E.: In Personal communications between Eric McDermid and David F. Manlove (2010)

    Google Scholar 

  12. McDermid, E., Irving, R.: Sex-equal stable matchings: complexity and exact algorithms. Algorithmica 68, 545–570 (2014)

    Article  MathSciNet  Google Scholar 

  13. Cseh, A., Manlove, D.F.: Stable marriage and roommates problems with restricted edges: complexity and approximability. Discret. Optim. 20, 62–89 (2016)

    Article  MathSciNet  Google Scholar 

  14. Mnich, M., Schlotter, I.: Stable marriage with covering constraints: a complete computational trichotomy (2016). CoRR, arXiv:1602.08230

  15. Kobayashi, H., Matsui, T.: Cheating strategies for the gale-shapley algorithm with complete preference lists. Algorithmica 58, 151–169 (2010)

    Article  MathSciNet  Google Scholar 

  16. Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)

    Book  Google Scholar 

  17. Downey R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)

    Google Scholar 

  18. Feder, T.: Stable networks and product graphs. Ph.D. thesis, Stanford University (1990)

    Google Scholar 

  19. Bredereck, R., Chen, J., Faliszewski, P., Guo, J., Niedermeier, R., Woeginger, G.J.: Parameterized algorithmics for computational social choice: nine research challenges (2014). CoRR, arXiv:1407.2143

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Gupta S., Roy, S.: Stable matching games: manipulation via subgraph isomorphism. In: Proceedings of the 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS) volume 65 of LIPIcs, pp. 29:1–29:14 (2016)

    Google Scholar 

  23. Gupta, S., Roy, S.: Stable matching games: manipulation via subgraph isomorphism. Algorithmica 10, 1–23 (2017)

    MATH  Google Scholar 

  24. Beyer, T., Hedetniemi, S.M.: Constant time generation of rooted trees. SIAM J. Comput. 9, 706–712 (1980)

    Article  MathSciNet  Google Scholar 

  25. Otter, Richard: The number of trees. Ann. Math. 49, 583–599 (1948)

    Article  MathSciNet  Google Scholar 

  26. Fomin, F. V., Lokshtanov, D., Panolan, F., Saurabh, S.: Representative sets of product families. J. ACM Trans. Algorithms, 13 (2017)

    Google Scholar 

  27. Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S., Rao, B.V.R.: Faster algorithms for finding and counting subgraphs. J. Comput. Syst. Sci. 78, 698–706 (2012)

    Article  MathSciNet  Google Scholar 

  28. Impagliazzo, R., Paturi, R.: The Complexity of k-SAT. In: The Proceedings of 14th IEEE Conference on Computational Complexity, pp. 237–240 (1999)

    Google Scholar 

  29. Adil, D., Gupta, S., Roy, S., Saurabh, S., Zehavi, M.: Parameterized algorithms for stable matching with ties and incomplete lists. Manuscript (2017)

    Google Scholar 

  30. Ronn, E.: NP-complete stable matching problem. J. Algorithms 11, 285–304 (1990)

    Article  MathSciNet  Google Scholar 

  31. Horton, J.D., Kilakos, K.: Minimum edge dominating sets. SIAM J. Discret. Math. 6, 375–387 (1993)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  33. Peters, D.: Graphical hedonic games of bounded treewidth. In: Proceedings of the 30th AAAI Conference on Artificial Intelligence, pp. 586–593 (2016)

    Google Scholar 

  34. Munera, D., Diaz, D., Abreu, S., Rossi, F., Saraswat, V., Codognet, P.: Solving hard stable matching problems via local search and cooperative parallelization. In: Proceedings of 29th AAAI Conference on Artificial Intelligence, pp. 1212–1218 (2015)

    Google Scholar 

  35. Gent I. P., Prosser, P: An empirical study of the stable marriage problem with ties and incomplete lists. In: Proceedings of the 15th European Conference on Artificial Intelligence, pp. 141–145. IOS Press (2002)

    Google Scholar 

  36. O’Malley, G.: Algorithmic aspects of stable matching problems. Ph.D. thesis, University of Glasgow (2007)

    Google Scholar 

  37. Chen, J., Hermelin, D., Sorge, M., Yedidsion, H.: How hard is it to satisfy (almost) all roommates? (2017). CoRR, arXiv:1707.04316

  38. Gupta, S., Roy, S., Saurabh, S., Zehavi, M., Balanced stable marriage: how close is close enough? (2017). CoRR, arXiv:1707.09545v1

  39. Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: On multiway cut parameterized above lower bounds. TOCT 5, 3:1–3:11 (2013)

    Article  MathSciNet  Google Scholar 

  40. Garg, S., Philip, G.: Raising the bar for vertex cover: fixed-parameter tractability above A higher guarantee. In: Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1152–1166 (2016)

    Google Scholar 

  41. Lokshtanov, D., Narayanaswamy, N.S., Raman, V., Ramanujan, M.S., Saurabh, S.: Faster parameterized algorithms using linear programming. ACM Trans. Algorithms 11, 15:1–15:31 (2014)

    Article  MathSciNet  Google Scholar 

  42. Raman, V., Ramanujan, M.S., Saurabh, S.: Paths, flowers and vertex cover. In: Proceedings of 19th Annual European Symposium of Algorothms (ESA), pp. 382–393 (2011)

    Chapter  Google Scholar 

  43. Gupta, S., Saurabh, S., Zehavi, M.: On treewidth and stable marriage (2017). CoRR, arXiv:1707.05404

Download references

Author information

Authors and Affiliations

  1. University of Bergen, Bergen, Norway

    Sushmita Gupta & Meirav Zehavi

  2. The Institute of Mathematical Sciences, HBNI, Chennai, India

    Sanjukta Roy & Saket Saurabh

Authors
  1. Sushmita Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sanjukta Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Saket Saurabh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Meirav Zehavi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saket Saurabh .

Editor information

Editors and Affiliations

  1. Indian Statistical Institute, New Delhi, India

    S.K. Neogy

  2. Indian Statistical Institute, New Delhi, India

    Ravindra B. Bapat

  3. Indian Statistical Institute, New Delhi, India

    Dipti Dubey

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Gupta, S., Roy, S., Saurabh, S., Zehavi, M. (2018). Some Hard Stable Marriage Problems: A Survey on Multivariate Analysis. In: Neogy, S., Bapat, R., Dubey, D. (eds) Mathematical Programming and Game Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-13-3059-9_8

Download citation

Publish with us

Policies and ethics

Search

Navigation