Our input is a bipartite graph G=(A\cup B,E) where each vertex in A\cup B has a preference list strictly ranking its neighbors. The vertices in A and in B are called students and courses, respectively. Each student a seeks to be matched to cap(a)\geq 1 many courses while each course b seeks cap(b)\geq 1 many students to be matched to it. The Gale-Shapley algorithm computes a pairwise-stable matching (one with no blocking edge) in G in linear time. We consider the problem of computing a popular matching in G - a matching M is popular if M cannot lose an election to any matching where vertices cast votes for one matching versus another. Our main contribution is to show that a max-size popular matching in G can be computed by the 2-level Gale-Shapley algorithm in linear time. This is an extension of the classical Gale-Shapley algorithm and we prove its correctness via linear programming.
@InProceedings{brandl_et_al:LIPIcs.FSTTCS.2017.19, author = {Brandl, Florian and Kavitha, Telikepalli}, title = {{Popular Matchings with Multiple Partners}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.19}, URN = {urn:nbn:de:0030-drops-83765}, doi = {10.4230/LIPIcs.FSTTCS.2017.19}, annote = {Keywords: Bipartite graphs, Linear programming duality, Gale-Shapley algorithm} }
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