Stable Marriage with Incomplete Lists and Ties

  • Kazuo Iwama
  • Shuichi Miyazaki
  • Yasufumi Morita
  • David Manlove
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1644)


The original stable marriage problem requires all men and women to submit a complete and strictly ordered preference list. This is obviously often unrealistic in practice, and several relaxations have been proposed, including the following two common ones: one is to allow an incomplete list, i.e., a man is permitted to accept only a subset of the women and vice versa. The other is to allow a preference list including ties. Fortunately, it is known that both relaxed problems can still be solved in polynomial time. In this paper, we show that the situation changes substantially if we allow both relaxations (incomplete lists and ties) at the same time: the problem not only becomes NP-hard, but also the optimal cost version has no approximation algorithm achieving the approximation ratio of N 1- , where N is the instance size, unless P=NP.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Gale and L.S. Shapley, “College admissions and the stability of marriage,” Amer. Math. Monthly, Vol.69, pp.9–15, 1962.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    D. Gale and M. Sotomayor, “Some remarks on the stable matching problem,” Discrete Applied Mathematics, Vol.11, pp.223–232, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    M.R. Garey and D.S. Johnson, “Computers and Intractability, A Guide to The Theory of NP-Completeness,” Freeman, San Francisco, 1979.zbMATHGoogle Scholar
  4. [4]
    D. Gusfield and R.W. Irving, “The Stable Marriage Problem: Structure and Algorithms,” MIT Press, Boston, MA, 1989.zbMATHGoogle Scholar
  5. [5]
    J. Håstad, “Clique is hard to approximate within n 1-ε, ” Proc. FOCS96, pp.627-636, 1996.Google Scholar
  6. [6]
    R.W. Irving, “Stable marriage and indifference,” Discrete Applied Mathematics, Vol.48, pp.261–272, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    T. Ohguro, “On the hardness of approximating minimization problems: in comparison with maximization classes,” Technical Report of IEICE, COMP94-112 pp.89–96, 1995.Google Scholar
  8. [8]
    E. Ronn, “NP-complete stable matching problems,” J. Algorithms, Vol.11, pp.285–304, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    M. Yannakakis and F. Gavril, “Edge dominating sets in graphs,” SIAM J. Applied Mathematics, Vol.18, No.1, pp.364–372, 1980.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Kazuo Iwama
    • 1
  • Shuichi Miyazaki
    • 1
  • Yasufumi Morita
    • 1
  • David Manlove
    • 2
  1. 1.School of InformaticsKyoto UniversityKyotoJapan
  2. 2.Dept. of Computing ScienceUniversity of GlasgowGlasgowScotland

Personalised recommendations