BIB-VERSION:: CS-TR-v2.0
ID:: ncstrl.dartmouthcs//TR2008-616
ENTRY:: April 16, 2008
ORGANIZATION:: Dartmouth College, Computer Science
REQUESTED-BY::  pw@cs.dartmouth.edu
REQUESTED-FOR:: villars@cs.dartmouth.edu
REQUESTED-DATE:: Sat Apr 12 08:06:39 EDT 2008
TITLE:: Bounded Unpopularity Matchings
TYPE:: Technical Report (paper)
REVISION:: 1
AUTHOR:: Huang, Chien-Chung
AUTHOR:: Telikepalli, Kavitha
AUTHOR:: Michail, Dimitrios
AUTHOR:: Nasre, Meghana
DATE:: April 2008
RETRIEVAL:: For a paper copy, email <reports@cs.dartmouth.edu>
RETRIEVAL:: For a paper copy, write to
       Technical Report Librarian
       Department of Computer Science
       Dartmouth College
       6211 Sudikoff Laboratory
       Hanover, NH 03755-3510
       USA
RETRIEVAL:: PDF at http://www.cs.dartmouth.edu/reports/TR2008-616.pdf
ABSTRACT:: 
       We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of \emph{popularity}. A matching $M$ is popular if there is no matching $M'$ such that more people prefer $M'$ to $M$ than the other way around. Determining whether a given instance admits a popular matching and, if so, finding  one, was studied in \cite{AIKM05}. If there is no popular matching, a reasonable substitute is a matching whose {\em unpopularity} is bounded. We consider two measures of unpopularity - {\em unpopularity factor} denoted by $u(M)$ and  {\em unpopularity margin} denoted by $g(M)$. McCutchen recently showed that computing a matching $M$
with the minimum value of $u(M)$ or $g(M)$ is NP-hard, and that if $G$ does not admit a popular matching, then we have $u(M) \ge 2$ for all matchings $M$ in $G$.
       Here we show that a matching $M$ that achieves $u(M) = 2$ can be computed in $O(m\sqrt{n})$ time (where $m$ is the number of edges in $G$ and $n$ is the number of nodes) provided a certain graph $H$ admits a matching that matches all people. We also describe a sequence of graphs: $H = H_2, H_3,\ldots,H_k$ such that if $H_k$ admits a matching that matches all people, then we can compute in $O(km\sqrt{n})$ time a matching $M$ such that $u(M) \le k-1$ and $g(M) \le n(1-\frac{2}{k})$. Simulation results suggest that our algorithm finds a matching with low unpopularity.
END:: ncstrl.dartmouthcs//TR2008-616