<XML><RECORDS><RECORD><REFERENCE_TYPE>10</REFERENCE_TYPE><REFNUM>9139</REFNUM><AUTHORS><AUTHOR>Biro,P.</AUTHOR><AUTHOR>Manlove,D.F.</AUTHOR><AUTHOR>Mittal,S.</AUTHOR></AUTHORS><YEAR>2009</YEAR><TITLE>Size versus stability in the Marriage problem</TITLE><PLACE_PUBLISHED>DCS Technical Report Series</PLACE_PUBLISHED><PUBLISHER>Dept of Computing Science, University of Glasgow</PUBLISHER><ISBN>TR-2009-297</ISBN><LABEL>Biro:2009:9139</LABEL><KEYWORDS><KEYWORD>Stable marriage problem; Stable matching; Blocking pair; Blocking agent; Inapproximability result; Polynomial-time algorithm</KEYWORD></KEYWORDS<ABSTRACT>Given an instance I of the classical Stable Marriage problem with Incomplete preference lists (SMI), a maximum cardinality matching can be larger than a stable matching. In many large-scale applications of SMI, we seek to match as many agents as possible. This motivates the problem of finding a maximum cardinality matching in I that admits the smallest number of blocking pairs (so is "as stable as possible"). We show that this problem is NP-hard and not approximable within n^{1-\varepsilon}, for any \varepsilon>0, unless P=NP, where n is the number of men in I. Further, even if all preference lists are of length at most 3, we show that the problem remains NP-hard and not approximable within \delta, for some \delta>1. By contrast, we give a polynomial-time algorithm for the case where the preference lists of one sex are of length at most 2. We also extend these results to the cases where (i) preference lists may include ties, and (ii) we seek to minimise the number of agents involved in a blocking pair.</ABSTRACT></RECORD></RECORDS></XML>