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Stable Marriage with Ties and Incomplete Lists

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

Synonyms

Stable matching problem

Years and Authors of Summarized Original Work

  • 2007; Iwama, Miyazaki, Yamauchi

Problem Definition

In the original setting of the stable marriage problem introduced by Gale and Shapley [2], each preference list has to include all members of the other party, and furthermore, each preference list must be totally ordered (see entry Stable Marriage also).

One natural extension of the problem is then to allow persons to include ties in preference lists. In this extension, there are three variants of the stability definition, super-stability, strong stability, and weak stability (see below for definitions). In the first two stability definitions, there are instances that admit no stable matching, but there is a polynomial-time algorithm in each case that determines if a given instance admits a stable matching and finds one if one exists [9]. On the other hand, in the case of weak stability, there always exists a stable matching, and one can be found in...

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Recommended Reading

  1. Canadian Resident Matching Service (CaRMS), http://www.carms.ca/

  2. Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69:9–15

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Gusfield D, Irving RW (1989) The stable marriage problem: structure and algorithms. MIT Press, Boston

    MATH  Google Scholar 

  5. Halldórsson MM, Irving RW, Iwama K, Manlove DF, Miyazaki S, Morita Y, Scott S (2003) Approximability results for stable marriage problems with ties. Theor Comput Sci 306:431–447

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Halldórsson MM, Iwama Ka, Miyazaki S, Yanagisawa H (2007) Improved approximation results of the stable marriage problem. ACM Trans Algorithms 3(3):Article No. 30

    Google Scholar 

  8. Huang C-C, Kavitha T (2014) An improved approximation algorithm for the stable marriage problem with one-sided ties. In: Proceedings of the IPCO 2014, Bonn. LNCS, vol 8494, pp 297–308

    Google Scholar 

  9. Irving RW (1994) Stable marriage and indifference. Discret Appl Math 48:261–272

    Article  MathSciNet  MATH  Google Scholar 

  10. Irving RW (1998) Matching medical students to pairs of hospitals: a new variation on a well-known theme. In: Proceedings of the ESA 1998, Venice. LNCS, vol 1461, pp 381–392

    Google Scholar 

  11. Irving RW, Manlove DF, Scott S (2000) The hospitals/residents problem with ties. In: Proceedings of the SWAT 2000, Bergen. LNCS, vol 1851, pp 259–271

    Google Scholar 

  12. Irving RW, Manlove DF, O’Malley G (2009) Stable marriage with ties and bounded length preference lists. J Discret Algorithms 7(2):213–219

    Article  MathSciNet  MATH  Google Scholar 

  13. Iwama K, Manlove DF, Miyazaki S, Morita Y (1999) Stable marriage with incomplete lists and ties. In: Proceedings of the ICALP 1999, Prague. LNCS, vol 1644, pp 443–452

    Google Scholar 

  14. Iwama K, Miyazaki S, Yamauchi N (2007) A 1.875-approximation algorithm for the stable marriage problem. In: Proceedings of the SODA 2007, New Orleans, pp 288–297

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Japanese Resident Matching Program (JRMP), http://www.jrmp.jp/

  17. Kavitha T, Mehlhorn K, Michail D, Paluch KE (2007) Strongly stable matchings in time O(nm) and extension to the hospitals-residents problem. ACM Trans Algorithms 3(2):Article No. 15

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Manlove DF (1999) Stable marriage with ties and unacceptable partners. Technical Report no. TR-1999-29 of the Computing Science Department of Glasgow University

    Google Scholar 

  21. Manlove DF, Irving RW, Iwama K, Miyazaki S, Morita Y (2002) Hard variants of stable marriage. Theor Comput Sci 276(1–2):261–279

    Article  MathSciNet  MATH  Google Scholar 

  22. McDermid EJ (2009) A 3/2-approximation algorithm for general stable marriage. In: Proceedings of the ICALP, Rhodes. LNCS, vol 5555, pp 689–700

    Google Scholar 

  23. Paluch KE (2014) Faster and simpler approximation of stable matchings. Algorithms 7(2):189–202

    Article  MathSciNet  Google Scholar 

  24. Yanagisawa H (2007) Approximation algorithms for stable marriage problems. Ph.D. Thesis, Kyoto University

    Google Scholar 

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Author information

Authors and Affiliations

  1. Computer Engineering, Kyoto University, Sakyo, Kyoto, Japan

    Kazuo Iwama

  2. School of Informatics, Kyoto University, Rm. 315, Engineering-Building No. 10, 606-8501, Sakyo, Kyoto, Japan

    Kazuo Iwama

  3. Academic Center for Computing and Media Studies, Kyoto University, Kyoto, Japan

    Shuichi Miyazaki

Authors
  1. Kazuo Iwama
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  2. Shuichi Miyazaki
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Corresponding author

Correspondence to Kazuo Iwama .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Iwama, K., Miyazaki, S. (2016). Stable Marriage with Ties and Incomplete Lists. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_805

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