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Popular Matchings with Two-Sided Preferences and One-Sided Ties

  • Ágnes CsehEmail author
  • Chien-Chung HuangEmail author
  • Telikepalli KavithaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)

Abstract

We are given a bipartite graph \(G = (A \cup B, E)\) where each vertex has a preference list ranking its neighbors: in particular, every \(a \in A\) ranks its neighbors in a strict order of preference, whereas the preference lists of \(b \in B\) may contain ties. A matching M is popular if there is no matching \(M'\) such that the number of vertices that prefer \(M'\) to M exceeds the number that prefer M to \(M'\). We show that the problem of deciding whether G admits a popular matching or not is \(\mathsf {NP}\)-hard. This is the case even when every \(b \in B\) either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each \(b \in B\) puts all its neighbors into a single tie. That is, all neighbors of b are tied in b’s list and and b desires to be matched to any of them. Our main result is an \(O(n^2)\) algorithm (where \(n = |A \cup B|\)) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in B have no preferences and do not care whether they are matched or not.

Keywords

Bipartite Graph Maximum Match Prefer Post Stable Match Edge Label 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.TU BerlinBerlinGermany
  2. 2.Chalmers UniversityGöteborgSweden
  3. 3.Tata Institute of Fundamental ResearchMumbaiIndia

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