Stabilization of Capacitated Matching Games

An edge-weighted, vertex-capacitated graph G is called stable if the value of a maximum-weight capacity-matching equals the value of a maximum-weight fractional capacity-matching. Stable graphs play a key role in characterizing the existence of stable solutions for popular combinatorial games that involve the structure of matchings in graphs, such as network bargaining games and cooperative matching games. The vertex-stabilizer problem asks to compute a minimum number of players to block (i.e., vertices of G to remove) in order to ensure stability for such games. The problem has been shown to be solvable in polynomial-time, for unit-capacity graphs. This stays true also if we impose the restriction that the set of players to block must not intersect with a given specified maximum matching of G. In this work, we investigate these algorithmic problems in the more general setting of arbitrary capacities. We show that the vertex-stabilizer problem with the additional restriction of avoiding a given maximum matching remains polynomial-time solvable. Differently, without this restriction, the vertex-stabilizer problem becomes NP-hard and even hard to approximate, in contrast to the unit-capacity case. Finally, in unit-capacity graphs there is an equivalence between the stability of a graph, existence of a stable solution for network bargaining games, and existence of a stable solution for cooperative matching games. We show that this equivalence does not extend to the capacitated case.Comment: 14 pages, 3 figure

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 . 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

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 . 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