Abstract
An edgeweighted graph G=(V,E) is called stable if the value of a maximumweight matching equals the value of a maximumweight 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 vertexremoval 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 polynomialtime 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 polynomialtime algorithm to compute a basic maximumweight 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 constantfactor 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.
BibTeX  Entry
@InProceedings{koh_et_al:LIPIcs:2018:9087,
author = {Zhuan Khye Koh and Laura Sanit{\`a}},
title = {{Stabilizing Weighted Graphs}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {83:183:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770767},
ISSN = {18688969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9087},
URN = {urn:nbn:de:0030drops90877},
doi = {10.4230/LIPIcs.ICALP.2018.83},
annote = {Keywords: combinatorial optimization, network bargaining, cooperative game}
}
Keywords: 

combinatorial optimization, network bargaining, cooperative game 
Collection: 

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) 
Issue Date: 

2018 
Date of publication: 

04.07.2018 