Finding Small Stabilizers for Unstable Graphs

  • Adrian Bock
  • Karthekeyan Chandrasekaran
  • Jochen Könemann
  • Britta Peis
  • Laura Sanità
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)


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.


Undirected Graph Regular Graph Vertex Cover Grand Coalition Maximum Match 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Adrian Bock
    • 1
  • Karthekeyan Chandrasekaran
    • 2
  • Jochen Könemann
    • 3
  • Britta Peis
    • 4
  • Laura Sanità
    • 3
  1. 1.EPFLSwitzerland
  2. 2.Harvard UniversityUSA
  3. 3.University of WaterlooCanada
  4. 4.RWTH Aachen UniversityGermany

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