Popular Half-Integral Matchings

In an instance G = (A∪B,E) of the stable marriage problem with strict and possibly incomplete preference lists, a matchingM is popular if there is no matchingM ′ where the vertices that prefer M ′ to M outnumber those that prefer M to M ′. All stable matchings are popular and there is a simple linear time algorithm to compute a maximum-size popular matching. More generally, what we seek is a min-cost popular matching where we assume there is a cost function c : E → Q. However there is no polynomial time algorithm currently known for solving this problem. Here we consider the following generalization of a popular matching called a popular half-integral matching: this is a fractional matching ~x = (M1 + M2)/2, where M1 and M2 are the 0-1 edge incidence vectors of matchings in G, such that ~x satisfies popularity constraints. We show that every popular half-integral matching is equivalent to a stable matching in a larger graph G∗. This allows us to solve the min-cost popular half-integral matching problem in polynomial time. 1998 ACM Subject Classification G.2.2 Graph algorithms, G.1.6 Linear programming