"Almost Stable" Matchings in the Roommates Problem

An instance of the classical Stable Roommates problem (sr) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an sr instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within n 1 2 −ε , for any ε > 0, unless P=NP. If the preference lists contain ties, we improve this result to n1−ε . Also, we show that, given an integer K and an sr instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the mi...

• Type: Conference
• Year: 2005
• Where: WAOA
• Authors: David J. Abraham, Péter Biró, David Manlove
• Keywords: Fewest Number, Preference Lists, Sr Instance, WAOA 2005