Efficient Algorithms for Weighted Rank-Maximal Matchings and Related Problems

  • Telikepalli Kavitha
  • Chintan D. Shah
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4288)


We consider the problem of designing efficient algorithms for computing certain matchings in a bipartite graph \(G =({\mathcal{A}} \cup {\mathcal{P}}, {\mathcal{E}})\), with a partition of the edge set as \({\mathcal{E}} = {\mathcal{E}}_1 {\mathbin {\dot{\cup}}} {\mathcal{E}}_2 \ldots {\mathbin {\dot{\cup}}} {\mathcal{E}}_r\). A matching is a set of (a, p) pairs, \(a \in {\mathcal{A}}, p\in{\mathcal{P}}\) such that each a and each p appears in at most one pair. We first consider the popular matching problem; an \(O(m\sqrt{n})\) algorithm to solve the popular matching problem was given in [3], where n is the number of vertices and m is the number of edges in the graph. Here we present an O(n ω ) randomized algorithm for this problem, where ω< 2.376 is the exponent of matrix multiplication. We next consider the rank-maximal matching problem; an \(O(\min(mn,Cm\sqrt{n}))\) algorithm was given in [7] for this problem. Here we give an O(Cn ω ) randomized algorithm, where C is the largest rank of an edge used in such a matching. We also consider a generalization of this problem, called the weighted rank-maximal matching problem, where vertices in \({\mathcal{A}}\) have positive weights.


Bipartite Graph Perfect Match Edge Incident Match Problem Maximum Rank 
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  1. 1.
    Abdulkadiroǧlu, A., Sönmez, T.: Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica 66(3), 689–701 (1998)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Abraham, D.J., Cechlárová, K., Manlove, D.F., Mehlhorn, K.: Pareto-optimality in house allocation problems. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 3–15. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Abraham, D.J., Irving, R.W., Kavitha, T., Mehlhorn, K.: Popular matchings. In: Proc. of 16th SODA, pp. 424–432 (2005)Google Scholar
  4. 4.
    Cheriyan, J.: Randomized Õ(M(|V|)) algorithms for problems in matching theory. SIAM Journal on Computing 26(6), 1635–1655 (1997)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hopcroft, J.E., Karp, R.M.: A n 5/2 Algorithm for Maximum Matchings in Bipartite Graphs. SIAM Journal on Computing 2, 225–231 (1973)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hylland, A., Zeckhauser, R.: The efficient allocation of individuals to positions. Journal of Political Economy 87(2), 293–314 (1979)CrossRefGoogle Scholar
  7. 7.
    Irving, R.W., Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: Rank-maximal matchings. In: Proc. of 15th SODA, pp. 68–75 (2004)Google Scholar
  8. 8.
    Lovász, L.: On determinants, matchings and random algorithms. Fundamentals of Computation Theory, 565–574 (1979)Google Scholar
  9. 9.
    Mucha, M., Sankowski, P.: Maximum Matchings via Gaussian Eliminatio. In: Proc. of 45th FOCS, pp. 248–255 (2004)Google Scholar
  10. 10.
    Mestre, J.: Weighted popular matchings. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 715–726. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Roth, A.E., Postlewaite, A.: Weak versus strong domination in a market with indivisible goods. Journal of Mathematical Economics 4, 131–137 (1977)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Yuan, Y.: Residence exchange wanted: a stable residence exchange problem. European Journal of Operational Research 90, 536–546 (1996)MATHCrossRefGoogle Scholar
  13. 13.
    Zhou, L.: On a conjecture by Gale about one-sided matching problems. Journal of Economic Theory 52(1), 123–135 (1990)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Telikepalli Kavitha
    • 1
  • Chintan D. Shah
    • 1
  1. 1.Indian Institute of ScienceBangaloreIndia

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