Finding small stabilizers for unstable graphs
An undirected graph is stable if the cardinality of a maximum matching equals the size of a minimum fractional vertex cover. We call a set of edges a stabilizer if its removal from yields a stable graph. In this paper we study the following natural edge-deletion question: given a graph , can we find a minimum-cardinality stabilizer? Stable graphs play an important role in cooperative game theory. In the classic matching game introduced by Shapley and Shubik (Int J Game Theory 1(1):111-130, 1971) we are given an undirected graph where vertices represent players, and we define the value of each subset as the cardinality of a maximum matching in the subgraph induced by . The core of such a game contains all fair allocations of the value of among the players, and is well-known to be non-empty iff graph is stable. The stabilizer problem addresses the question of how to modify the graph to ensure that the core is non-empty. We show that this problem is vertex-cover hard. We prove that every minimum-cardinality stabilizer avoids some maximum matching of . We use this insight to give efficient approximation algorithms for sparse graphs and for regular graphs.