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∗(kn) 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∗(2n) time algorithm for the truly popular matching problem. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms

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