9 research outputs found
Popular Matchings in Complete Graphs
Our input is a complete graph on vertices where each vertex
has a strict ranking of all other vertices in . Our goal is to construct a
matching in that is popular. A matching is popular if does not lose
a head-to-head election against any matching , where each vertex casts a
vote for the matching in where it gets assigned a better partner.
The popular matching problem is to decide whether a popular matching exists or
not. The popular matching problem in is easy to solve for odd .
Surprisingly, the problem becomes NP-hard for even , as we show here.Comment: Appeared at FSTTCS 201
Popular Roommates in Simply Exponential Time
We consider the popular matching problem in a graph G = (V,E) on n vertices with strict preferences. A matching M is popular if there is no matching N in G such that vertices that prefer N to M outnumber those that prefer M to N. It is known that it is NP-hard to decide if G has a popular matching or not. There is no faster algorithm known for this problem than the brute force algorithm that could take n! time. Here we show a simply exponential time algorithm for this problem, i.e., one that runs in O^*(k^n) time, where k is a constant.
We use the recent breakthrough result on the maximum number of stable matchings possible in such instances to analyze our algorithm for the popular matching problem. We identify a natural (also, hard) subclass of popular matchings called truly popular matchings and show an O^*(2^n) time algorithm for the truly popular matching problem
Popular Matchings in Complete Graphs
Our input is a complete graph G = (V,E) on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is "globally stable" or popular. A matching M is popular if M does not lose a head-to-head election against any matching M\u27: here each vertex casts a vote for the matching in {M,M\u27} where it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes NP-hard for even n, as we show here