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Stabilizing Weighted Graphs

Authors Zhuan Khye Koh, Laura Sanità



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Zhuan Khye Koh
  • Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada
Laura Sanità
  • Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada

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Zhuan Khye Koh and Laura Sanità. Stabilizing Weighted Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 83:1-83:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.83

Abstract

An edge-weighted graph G=(V,E) 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 some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the literature so far only hold for unweighted instances, i.e., assuming unit weights on the edges of G. 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 particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with minimum number of odd cycles in its support. This generalizes a fundamental and classical result on unweighted matchings given by Balas more than 30 years ago, which we expect to prove useful beyond this particular application. 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(Delta)-approximation algorithm for the problem, where Delta is the maximum degree of a node in G.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Matchings and factors
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Discrete optimization
  • Theory of computation → Algorithmic game theory
  • Theory of computation → Network games
Keywords
  • combinatorial optimization
  • network bargaining
  • cooperative game

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