Improving Man-Optimal Stable Matchings by Minimum Change of Preference Lists

Abstract

In the stable marriage problem, any instance admits the so-called man-optimal stable matching, in which every man is assigned the best possible partner. However, there are instances for which all men receive low-ranked partners even in the man-optimal stable matching. In this paper we consider the problem of improving the man-optimal stable matching by changing only one man’s preference list. We show that the optimization variant and the decision variant of this problem can be solved in time O(n3) and O(n2), respectively, where n is the number of men (women) in an input. We further extend the problem so that we are allowed to change k men’s preference lists. We show that the problem is W[1]-hard with respect to the parameter k and give O(n2k+1)-time and O(nk+1)-time exact algorithms for the optimization and decision variants, respectively. Finally, we show that the problems become easy when k = n; we give O(n2.5 log n)-time and O(n2)-time algorithms for the optimization and decision variants, respectively.

Keywords

• combinatorial optimization
• polynomial-time algorithms
• stable marriage problem
• man-optimal stable matching