# Dynamic rank-maximal and popular matchings

Ontology type: schema:ScholarlyArticle

### Article Info

DATE

2018-09-21

AUTHORS 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=(A∪P,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(\mathcal {A}\cup \mathcal {P},E)$$\end{document}, where A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} denotes a set of applicants, P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {P}$$\end{document} 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))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(r(m+n))$$\end{document}-time algorithm to update an existing rank-maximal matching under each of these changes. When r=o(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=o(n)$$\end{document}, 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,rn)m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\min ((r+n,r\sqrt{n})m)$$\end{document}. Our algorithm can also be used for maintaining a popular matching in the one-sided preference model in O(m+n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(m+n)$$\end{document} time, whenever one exists. More... »

PAGES

523-545

### References to SciGraph publications

• 2014-11-08. Rank-Maximal Matchings – Structure and Algorithms in ALGORITHMS AND COMPUTATION
• 2006. Dynamic Matching Markets and Voting Paths in ALGORITHM THEORY – SWAT 2006
• 2013. Capacitated Rank-Maximal Matchings in ALGORITHMS AND COMPLEXITY
• 2004. Pareto Optimality in House Allocation Problems in ALGORITHMS AND COMPUTATION
• 2006. Efficient Algorithms for Weighted Rank-Maximal Matchings and Related Problems in ALGORITHMS AND COMPUTATION
• 2015-06-20. Maintaining Near-Popular Matchings in AUTOMATA, LANGUAGES, AND PROGRAMMING
• ### Journal

TITLE

Journal of Combinatorial Optimization

ISSUE

2

VOLUME

37

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s10878-018-0348-9

DOI

http://dx.doi.org/10.1007/s10878-018-0348-9

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1107185913

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