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. [13], which takes time \(O(\min ((r+n,r\sqrt{n})m)\).
R.V. ArvindāPart of the work was done when the author was a summer intern at Chennai Mathematical Institute.
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Notes
- 1.
In Irving et al.ās 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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Acknowledgement
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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Nimbhorkar, P., Rameshwar V., A. (2017). Dynamic Rank-Maximal Matchings. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_36
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