Theory and Methodology A new Lagrangean relaxation approach for the hop-constrained minimum spanning tree problem Lu  õs Gouveia a, * , Cristina Requejo b a DEIO/CIO, Faculdade de Ci^ encias da Universidade de Lisboa, Bloco C2, Campo Grande, 1749-016 Lisboa, Portugal b Departamento de Matem atica/Instituto de Telecomunicac ß~ oes, Faculdade de Ci^ encias e Tecnologia da Universidade de Coimbra, 3000 Coimbra, Portugal Received 19 July 1999; accepted 10 May 2000 Abstract We present a new Lagrangean relaxation for the hop-constrained minimum spanning tree problem HMST). The HMST is NP-hard and models the design of centralized telecommunication networks with quality of service con- straints.ThelinearprogrammingLP)relaxationofahop-indexed¯ow-basedmodelrecentlypresentedintheliterature see Gouveia, L., 1998. Using variable rede®nition for computing lower bounds for minimum spanning and Steiner trees with hop constraints. INFORMS Journal on Computing 10, 180±188) produces very tight bounds but has the disadvantage of being very time consuming, especially for dense graphs. In this paper, we present a new Lagrangean relaxation which is derived from the hop-indexed ¯ow based formulation. Our computational results indicate that the lower bounds given by the new relaxation dominate the lower bounds given by previous Lagrangean relaxations. Our results also show that for dense graphs the new Lagrangean relaxation proves to be a reasonable alternative to solving the LP relaxation of the hop-indexed model. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Design of centralized networks; Quality of service constraints; Hop constraints; Lagrangean relaxation; Multicommodity ¯ows 1. Introduction The hop-constrained minimum spanning tree problem HMST) is de®ned as follows: Consider a graph G N; E with N f0; 1; ... ; ng, costs c e for each edge e 2 E and a natural number H .Wewantto®nd the minimum spanning tree T such that the number of hops arcs) in the unique path from the root node, node 0, to any other node is not greater than H . European Journal of Operational Research 132 2001) 539±552 www.elsevier.com/locate/dsw * Corresponding author. Tel.: +351-217500409; fax: +351-217500081. E-mail address: lgouveia@fc.ul.pt L. Gouveia). 0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII:S0377-221700)00143-0