Theory and Methodology A quadratic programming approach to the determination of an upper bound on the weighted stability number Carlos J. Luz a, * , Domingos M. Cardoso b a Escola Superior de Tecnologia, Instituto Polit ecnico de Set  ubal, Rua Vale de Chaves, Estefanilha, 2914-508 Set  ubal, Portugal b Departamento de Matem atica, Universidade de Aveiro, 3810-193 Aveiro, Portugal Received 5 August 1999; accepted 16 May 2000 Abstract Inapreviouswork,theauthorshaveintroducedanupperboundonthestabilitynumberofagraphandseveralofits propertiesweregiven.Thedeterminationofthisupperboundwasdonebyaquadraticprogrammingapproachwhose implementation has given good computational results. We now extend this bound to the weighted case, i.e., an upper bound on the weighted stability of an arbitrary graph with node weights is presented. Similarly to the non-weighted case,thededucedboundallowsustogiveanecessaryandsucientconditiontoaweightedgraphthatattainsthegiven bound. Also a heuristic for determining the maximum weight stable set is proposed which is based on an integrality property of a convex quadratic problem that produces the bound. Some comments about the proposed approach will conclude the paper. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Graph theory; Weighted stability number; Quadratic programming 1. Introduction Let G V ; E beasimplegraphwhere V fv 1 ; ... ; v n g and E denote,respectively,thevertexandedge sets. Throughout of the paper we will suppose that E is non-empty and consider, assigned to each vertex v i 2 V ; a positive weight w i 2 R  . A stable set independent set) of G is a subset of nodes of V whose el- ements are pairwise non-adjacent. The stability number independence number) of G is de®ned as the cardinalityofthelargeststablesetandisusuallydenotedby aG: Amaximumweightstablesetof G isa stable set for which the sum of vertex weights is maximum. This maximum sum is referred to as the weighted stability number of G and will be denoted by a w G: The problem of ®nding aG and conse- quently a w G; asthisnumberequals aG intheparticularcasewhereallvertexweightsareequalto1)is European Journal of Operational Research 132 2001) 569±581 www.elsevier.com/locate/dsw * Corresponding author. Tel.: +351-265-790018; fax: +351-265-72-1869. E-mail addresses: cluz@est.ips.pt C.J. Luz), dcardoso@mat.ua.pt D.M. Cardoso). 0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII:S0377-221700)00162-4