Popular Roommates in Simply Exponential Time

Author Telikepalli Kavitha

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Telikepalli Kavitha
  • Tata Institute of Fundamental Research, Mumbai, India


Work done while visiting Max-Planck-Institut für Informatik, Saarland Informatics Campus, Germany. Thanks to Neeldhara Misra for asking me about fast exponential time algorithms for the popular roommates problem.

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Telikepalli Kavitha. Popular Roommates in Simply Exponential Time. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Roommates instance
  • Popular matching
  • Stable matching
  • Dual certificate


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