Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T19:22:44.539Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Likely Number of Solutions for the Stable Marriage Problem

Published online by Cambridge University Press:  01 May 2009

CRAIG LENNON  and
BORIS PITTEL
CRAIG LENNON
Affiliation:
Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210, USA (e-mail: [email protected], [email protected])
BORIS PITTEL
Affiliation:
Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210, USA (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

An instance of a size-n stable marriage problem involves n men and n women, each individually ranking all members of opposite sex in order of preference as a potential marriage partner. A complete matching, a set of n marriages, is called stable if no unmatched man and woman prefer each other to their partners in the matching. It is known that, for every instance of marriage partner preferences, there exists at least one stable matching, and that there are instances with exponentially many stable matchings. Our focus is on a random instance chosen uniformly from among all (n!)2n possible instances. The second author had proved that the expected number of stable marriages is of order nlnn, while its likely value is of order n1/2−o(1) at least. In this paper the second moment of that number is shown to be of order (nlnn)2. The combination of the two moment estimates implies that the fraction of problem instances with roughly cnlnn solutions is at least 0.84. Whether this fraction is asymptotic to 1 remains an open question.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 3 , May 2009 , pp. 371 - 421
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Billingsley, P. (1995) Probability and Measure, 3rd edn, Wiley.Google Scholar
[2]Durrett, R. (2005) Probability: Theory and Examples, 3rd edn, Brooks/Cole.Google Scholar
[3]Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn, Wiley.Google Scholar
[4]Gale, D. and Shapley, L. S. (1962) College admissions and the stability of marriage. Amer. Math. Monthly 69 9–15.Google Scholar
[5]Gusfield, D. and Irving, R. W. (1989) The Stable Marriage Problem: Structure and Algorithms, MIT Press.Google Scholar
[6]Irving, R. W. and Leather, P. (1986) The complexity of counting stable marriages. SIAM J. Comput. 15 655–667.CrossRefGoogle Scholar
[7]Knuth, D. E. (1976) Mariages Stables et Leurs Relations avec d' Autres Problémes Combinatoires, Les Presses de l'Université de Montréal. Translation by Martin Goldstein: Stable Marriage and its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms, AMS (1997).Google Scholar
[8]Knuth, D. E. (1988) Personal communication.Google Scholar
[9]Knuth, D. E., Motwani, R. and Pittel, B. (1990) Stable husbands. Random Struct. Alg. 1 1–14.Google Scholar
[10]Lennon, C. (2007) On the likely number of stable marriages. PhD dissertation, The Ohio State University, available at www.ohiolink.edu.Google Scholar
[11]McVitie, D. G. and Wilson, L. B. (1971) The stable marriage problem. Comm. Assoc. Comput. Mach. 14 486–490.Google Scholar
[12]Mertens, S. (2005) Random stable matchings. J. Statist. Mech. Theory Exp. 10 1–12.Google Scholar
[13]Pittel, B. (1989) The average number of stable matchings. SIAM J. Discrete Math. 2 520–549.CrossRefGoogle Scholar
[14]Pittel, B. (1991) On a random instance of the ‘stable roommates’ problem: Likely behavior of the proposal algorithm. Combin. Probab. Comput. 2 53–92.CrossRefGoogle Scholar
[15]Pittel, B. (1992) On likely solutions of a stable marriage problem. Ann. Appl. Probab. 2 358–401.CrossRefGoogle Scholar
[16]Pittel, B. (1993) The ‘stable roommates’ problem with random preferences. Ann. Probab. 21 1441–1477.Google Scholar
[17]Pittel, B. and Irving, R. W. (1994) An upper bound for the solvability probability of a random stable roommates instance. Random Struct. Alg. 5 465–486.CrossRefGoogle Scholar
[18]Wilson, L. B. (1972) An analysis of the stable marriage assignment problem. BIT 12 569–575.CrossRefGoogle Scholar