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Fair Matchings and Related Problems

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Abstract

Let \(G = (A \cup B, E)\) be a bipartite graph, where every vertex ranks its neighbors in an order of preference (with ties allowed) and let \(r\) be the worst rank used. A matching \(M\) is fair in \(G\) if it has maximum cardinality, subject to this, \(M\) matches the minimum number of vertices to rank \(r\) neighbors, subject to that, \(M\) matches the minimum number of vertices to rank \((r-1)\) neighbors, and so on. We show an efficient combinatorial algorithm based on LP duality to compute a fair matching in \(G\). We also show a scaling based algorithm for the fair b-matching problem. Our two algorithms can be extended to solve other profile-based matching problems. In designing our combinatorial algorithm, we show how to solve a generalized version of the minimum weighted vertex cover problem in bipartite graphs, using a single-source shortest paths computation—this can be of independent interest.

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Notes

  1. We note that one can easily compute the value of \(r^*\) in \(\tilde{O}(r^{*}m\sqrt{n})\) or in \(\tilde{O}(r^{*}n^{\omega })\) time with high probability (see Sect. 2.2 for details). Thus the brute-force reduction to weighted matching can be improved to have running times of \(O(r^*mn)\) or \(\tilde{O}((r^*)^2 m\sqrt{n})\).

  2. Sng used the term “generous maximum matching”.

  3. Irving originally called it the “greedy matching” problem.

  4. Let \(\pi \) be the potential function and let \(C\) be any directed cycle in the residual graph to the 1-optimal flow. Then \(\sum _{e \in C} c_e = \sum _{e \in C} c^{\pi }_e \ge n \cdot (-1)\). Since edge costs are multiples of \(2^{1 + \lceil \log n \rceil }\), the sums must be nonnegative and hence there is no negative cost cycle in the residual graph. This implies optimality.

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Authors and Affiliations

  1. Chalmers University of Technology, Gothenburg, Sweden

    Chien-Chung Huang

  2. Tata Institute of Fundamental Research, Mumbai, India

    Telikepalli Kavitha

  3. Max-Planck Institut für Informatik, Saarbrücken, Germany

    Kurt Mehlhorn

  4. Harokopio University, Athens, Greece

    Dimitrios Michail

Authors
  1. Chien-Chung Huang
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  2. Telikepalli Kavitha
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  3. Kurt Mehlhorn
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  4. Dimitrios Michail
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Corresponding author

Correspondence to Chien-Chung Huang.

Additional information

This work is based on two pre-prints [1,2]. A preliminary version of this work appeared in the 33rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pages 339–350, 2013.

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Huang, CC., Kavitha, T., Mehlhorn, K. et al. Fair Matchings and Related Problems. Algorithmica 74, 1184–1203 (2016). https://doi.org/10.1007/s00453-015-9994-9

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