Stabilizing weighted graphs

Zhuan Khye Koh, Laura Sanità

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

An edge-weighted graph G is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in network bargaining games and cooperative matching games, because they characterize instances that admit stable outcomes. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P = NP. In this setting, we develop an O(Δ)-approximation algorithm for the problem, where Δ is the maximum degree of a node in G.
Original languageEnglish
Pages (from-to)1318-1341
Number of pages24
JournalMathematics of Operations Research
Volume45
Issue number4
DOIs
Publication statusPublished - 1 Nov 2020

Keywords

  • Game theory
  • Matching
  • Network bargaining

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