Finding small stabilizers for unstable graphs

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

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)173-196
Number of pages24
JournalMathematical Programming
Issue number1-2
StatePublished - Dec 1 2015


  • Game theory
  • Matchings
  • Network bargaining

ASJC Scopus subject areas

  • Software
  • Mathematics(all)


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