On solution concepts for matching games

A matching game is a cooperative game (N,v) defined on a graph G = (N,E) with an edge weighting w:E→R+ . The player set is N and the value of a coalition S ⊆ N is defined as the maximum weight of a matching in the subgraph induced by S. First we present an O(nm + n 2logn) algorithm that tests if the core of a matching game defined on a weighted graph with n vertices and m edges is nonempty and that computes a core allocation if the core is nonempty. This improves previous work based on the ellipsoid method. Second we show that the nucleolus of an n-player matching game with nonempty core can be computed in O(n 4) time. This generalizes the corresponding result of Solymosi and Raghavan for assignment games. Third we show that determining an imputation with minimum number of blocking pairs is an NP-hard problem, even for matching games with unit edge weights