Abstract
We consider the problem of matching applicants to posts where applicants have preferences over posts. Thus the input to our problem is a bipartite graph \(G=(\mathcal {A}\cup \mathcal {P},E)\), where \(\mathcal {A}\) denotes a set of applicants, \(\mathcal {P}\) is a set of posts, and there are ranks on edges which denote the preferences of applicants over posts. A matching M in G is called rank-maximal if it matches the maximum number of applicants to their rank 1 posts, subject to this the maximum number of applicants to their rank 2 posts, and so on. We consider this problem in a dynamic setting, where vertices and edges can be added and deleted at any point. Let n and m be the number of vertices and edges in an instance G, and r be the maximum rank used by any rank-maximal matching in G. We give a simple \(O(r(m+n))\)-time algorithm to update an existing rank-maximal matching under each of these changes. When \(r=o(n)\), this is faster than recomputing a rank-maximal matching completely using a known algorithm like that of Irving et al. (ACM Trans Algorithms 2(4):602–610, 2006), which takes time \(O(\min ((r+n,r\sqrt{n})m)\). Our algorithm can also be used for maintaining a popular matching in the one-sided preference model in \(O(m+n)\) time, whenever one exists.
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Notes
In Irving et al.’s algorithm, these labels are called \(\mathcal {E}_1,\mathcal {O}_1,\mathcal {U}_1\). We omit the subscripts for the sake of bravity. The subscripts are clear from the stage under consideration.
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Acknowledgements
We thank anonymous reviewers for their comments on an earlier version of this paper. We thank Meghana Nasre for helpful discussions.
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P. Nimbhorkar: Partially supported by Infosys grant and Tata Trust. V. A. Rameshwar: Part of the work was done when the author was a student at Birla Institute of Technology and Science, Pilani, Hyderabad Campus and a summer intern at Chennai Mathematical Institute.
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Nimbhorkar, P., Rameshwar, V.A. Dynamic rank-maximal and popular matchings. J Comb Optim 37, 523–545 (2019). https://doi.org/10.1007/s10878-018-0348-9
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DOI: https://doi.org/10.1007/s10878-018-0348-9