An undirected graph (Formula presented.) is stable if the cardinality of a maximum matching equals the size of a minimum fractional vertex cover. We call a set of edges (Formula presented.) a stabilizer if its removal from (Formula presented.) yields a stable graph. In this paper we study the following natural edge-deletion question: given a graph (Formula presented.), 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 (Formula presented.) where vertices represent players, and we define the value of each subset (Formula presented.) as the cardinality of a maximum matching in the subgraph induced by (Formula presented.). The core of such a game contains all fair allocations of the value of \$\$V\$\$V among the players, and is well-known to be non-empty iff graph \$\$G\$\$G 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 \$\$G\$\$G. We use this insight to give efficient approximation algorithms for sparse graphs and for regular graphs.

### Finding small stabilizers for unstable graphs

#### Abstract

An undirected graph (Formula presented.) is stable if the cardinality of a maximum matching equals the size of a minimum fractional vertex cover. We call a set of edges (Formula presented.) a stabilizer if its removal from (Formula presented.) yields a stable graph. In this paper we study the following natural edge-deletion question: given a graph (Formula presented.), 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 (Formula presented.) where vertices represent players, and we define the value of each subset (Formula presented.) as the cardinality of a maximum matching in the subgraph induced by (Formula presented.). The core of such a game contains all fair allocations of the value of \$\$V\$\$V among the players, and is well-known to be non-empty iff graph \$\$G\$\$G 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 \$\$G\$\$G. We use this insight to give efficient approximation algorithms for sparse graphs and for regular graphs.
##### Scheda breve Scheda completa Scheda completa (DC)
2015
2014
Bock, Adrian; Chandrasekaran, Karthekeyan; Konemann, Jochen; Peis, Britta; Sanità, Laura
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11565/4063840`
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