# Finding small stabilizers for unstable graphs

Adrian Bock, Karthekeyan Chandrasekaran, Jochen Könemann, Britta Peis, Laura Sanità

1 Citaat (Scopus)

## Samenvatting

An undirected graph G = (V,E) is stable if its inessential vertices (those that are exposed by at least one maximum matching) form a stable set. We call a set of edges F ⊆ E a stabilizer if its removal from G yields a stable graph. In this paper we study the following natural edge-deletion question: given a graph G = (V,E), 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 [19] we are given an undirected graph G = (V,E) where vertices represent players, and we define the value of each subset S ⊆ V as the cardinality of a maximum matching in the subgraph induced by S. The core of such a game contains all fair allocations of the value of V among the players, and is well-known to be non-empty iff graph 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 then prove that there is a minimum-cardinality stabilizer that avoids some maximum matching of G. We use this insight to give efficient approximation algorithms for sparse graphs and for regular graphs.

Originele taal-2 Engels Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Proceedings Springer 150-161 12 9783319075563 https://doi.org/10.1007/978-3-319-07557-0_13 Gepubliceerd - 2014 17th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2014 - Bonn, DuitslandDuur: 23 jun. 2014 → 25 jun. 2014

### Publicatie series

Naam Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 8494 LNCS 0302-9743 1611-3349

### Congres

Congres 17th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2014 Duitsland Bonn 23/06/14 → 25/06/14

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