Summary: A Well­Behaved Extension
of the Vertex Covering Problem
A. A. Ageev \Lambda
Abstract
The paper shows that several well­known results on properties of optimal solu­
tions of the minimum weight vertex covering problem (Balinski [1], Balinski and
Spielberg [2], Nemhauser and Trotter [8], Hammer et al.[7], Bourjolly et al.[3])
remain true for an extension of it.
1 Introduction
Let G = (V; E) be a undirected graph with vertex set V and edge set E. A subset X ` V
is called a vertex covering if every edge of G has at least one endpoint in X. The minimum
weight vertex covering problem (VCP) is, given a graph G with positive vertex weights
c i (i 2 V ), to find a vertex covering X \Lambda with the minimum weight
P
i2X \Lambda c i . Clearly, a
set X ` V is a vertex covering if and only if its complement V n X consists of pairwise
nonadjacent vertices, i.e., is stable. So finding a minimum weight vertex covering is
equivalent to finding a maximum weight stable set. Both problems are classic in discrete
optimization and have been extensively investigated for recent decades. Despite NP­
hardness (see Garey and Johnson [5]) they enjoy a number of remarkable properties

Source: Ageev, Alexandr - Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk

Collections: Mathematics