Advertisement

Counting Popular Matchings in House Allocation Problems

  • Rupam Acharyya
  • Sourav Chakraborty
  • Nitesh Jha
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8476)

Abstract

We study the problem of counting the number of popular matchings in a given instance. McDermid and Irving gave a poly-time algorithm for counting the number of popular matchings when the preference lists are strictly ordered. We first consider the case of ties in preference lists. Nasre proved that the problem of counting the number of popular matching is #P-hard when there are ties. We give an FPRAS for this problem.

We then consider the popular matching problem where preference lists are strictly ordered but each house has a capacity associated with it. We give a switching graph characterization of popular matchings in this case. Such characterizations were studied earlier for the case of strictly ordered preference lists (McDermid and Irving) and for preference lists with ties (Nasre). We use our characterization to prove that counting popular matchings in capacitated case is #P-hard.

Keywords

Bipartite Graph Outgoing Edge Maximum Match Stable Matchings Incoming Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. SIAM J. Comput. 37(4), 1030–1045 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bhatnagar, N., Greenberg, S., Randall, D.: Sampling stable marriages: why spouse-swapping won’t work. In: SODA, pp. 1223–1232 (2008)Google Scholar
  3. 3.
    Chebolu, P., Goldberg, L.A., Martin, R.A.: The complexity of approximately counting stable matchings. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010. LNCS, vol. 6302, pp. 81–94. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Gale, D., Shapley, L.S.: College admissions and the stability of marriage. The American Mathematical Monthly 69(1), 9–15 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Gärdenfors, P.: Match making: assignments based on bilateral preferences. Behavioral Science 20(3), 166–173 (1975)CrossRefGoogle Scholar
  6. 6.
    Irving, R.W., Leather, P.: The complexity of counting stable marriages. SIAM J. Comput. 15(3), 655–667 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. In: STOC, pp. 712–721 (2001)Google Scholar
  8. 8.
    Kavitha, T., Mestre, J., Nasre, M.: Popular mixed matchings. Theor. Comput. Sci. 412(24), 2679–2690 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Lovász, L., Plummer, M.D.: Matching theory. North-Holland Mathematics Studies, vol. 121. North-Holland Publishing Co., Amsterdam (1986), Annals of Discrete Mathematics, 29Google Scholar
  10. 10.
    Mahdian, M.: Random popular matchings. In: ACM Conference on Electronic Commerce, pp. 238–242 (2006)Google Scholar
  11. 11.
    McCutchen, R.M.: The least-unpopularity-factor and least-unpopularity-margin criteria for matching problems with one-sided preferences. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 593–604. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    McDermid, E., Irving, R.W.: Popular matchings: structure and algorithms. J. Comb. Optim. 22(3), 339–358 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Nasre, M.: Popular matchings: Structure and cheating strategies. In: STACS, pp. 412–423 (2013)Google Scholar
  14. 14.
    Sng, C.T.S., Manlove, D.: Popular matchings in the weighted capacitated house allocation problem. J. Discrete Algorithms 8(2), 102–116 (2010)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Rupam Acharyya
    • 1
  • Sourav Chakraborty
    • 1
  • Nitesh Jha
    • 1
  1. 1.Chennai Mathematical InstituteChennaiIndia

Personalised recommendations