MPI-INF/SWS Research Reports 1991-2021

# MPI-I-2010-1-001

## Maximum cardinality popular matchings in strict two-sided preference lists

### Huang, Chien-Chung and Kavitha, Telikepalli

#### November 2010, 17 pages.

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##### Status: available - back from printing

We consider the problem of computing a maximum cardinality {\em popular} matching in a bipartite graph $G = (\A\cup\B, E)$ where each vertex $u \in \A\cup\B$ ranks its neighbors in a strict order of preference. This is the same as an instance of the {\em stable marriage} problem with incomplete lists. A matching $M^*$ is said to be popular if there is no matching $M$ such that more vertices are better off in $M$ than in $M^*$. \smallskip Popular matchings have been extensively studied in the case of one-sided preference lists, i.e., only vertices of $\A$ have preferences over their neighbors while vertices in $\B$ have no preferences; polynomial time algorithms have been shown here to determine if a given instance admits a popular matching or not and if so, to compute one with maximum cardinality. It has very recently been shown that for two-sided preference lists, the problem of determining if a given instance admits a popular matching or not is NP-complete. However this hardness result assumes that preference lists have {\em ties}. When preference lists are {\em strict}, it is easy to show that popular matchings always exist since stable matchings always exist and they are popular. But the complexity of computing a maximum cardinality popular matching was unknown. In this paper we show an $O(mn)$ algorithm for this problem, where $n = |\A| + |\B|$ and $m = |E|$.

URL to this document: https://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/2010-1-001

BibTeX
@TECHREPORT{HuangKavitha2010,
AUTHOR = {Huang, Chien-Chung and Kavitha, Telikepalli},
TITLE = {Maximum cardinality popular matchings in strict two-sided preference lists},
TYPE = {Research Report},
INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
ADDRESS = {Stuhlsatzenhausweg 85, 66123 Saarbr{\"u}cken, Germany},
NUMBER = {MPI-I-2010-1-001},
MONTH = {November},
YEAR = {2010},
ISSN = {0946-011X},
}