Stabilizing Weighted Graphs

TL;DR

This paper introduces the first polynomial-time algorithm for transforming weighted graphs into stable graphs by vertex removal, extending previous unweighted results and providing new structural insights into fractional matchings.

Contribution

It presents a novel polynomial-time algorithm for vertex removal to achieve stability in weighted graphs, generalizing classical unweighted matching results and developing new structural properties.

Findings

First polynomial-time algorithm for weighted graphs

Development of a minimum odd cycle fractional matching algorithm

NP-hardness of edge removal for stability with approximation algorithm

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 $G$.