Stabilizing Weighted Graphs
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Date
2017-08-29
Authors
Koh, Zhuan Khye
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
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 may be 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(\Delta )-approximation algorithm for the problem, where \Delta is
the maximum degree of a node in G.
Description
Keywords
Matching, Game Theory, Network Bargaining, Approximation Algorithm