A new Lagrangean relaxation approach for the hop-constrained minimum spanning tree problem

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 constraints. The linear programming (LP) relaxation of a hop-indexed¯ow-based model recently presented in the literature (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

Analysis of Practical Aspects of Multi-Plane Routing-Based Load Balancing Approach for Future Link-State Convergent All-IP Access Networks

“…We use hop-constraint to select an optimal set of RPs that meet our RP-construction objectives while avoiding redundant paths. We have adopted the hopconstraint optimization which was originally introduced in [27] for spanning tree constructions. Here, we have added more constraints and have reformulated certain representations in order to adjust the optimization problem to our work.…”

A Multi-Population Genetic Algorithm for Tree-Shaped Network Design Problems

“…Let H be the pre-defined maximum number of arcs in any path connecting the source vertex to any other vertex. Several formulations have been provided by Gouveia and his co-authors, see (Gouveia and Requejo, 2001;Gouveia et al, 2008) and the references therein.…”

“…The server wishes to transmit identical information to all clients, and does so by transmitting the data to the vertices it directly connects to, and these latter vertices forward incoming data to their respective children in the tree. The limit on the number of arcs in each path is usually used to guarantee a certain quality of service with respect to availability and reliability constraints (LeBlanc and Reddoch, 1990;Woolston and Albin, 1988), as well as lower delays (Gouveia and Requejo, 2001), since they limit the number of arcs in each path from the central service provider. (Woolston and Albin, 1988) have shown that, by including this type of constraints it is possible to generate network designs with a much better quality of service and with only a marginal increase in the total cost.…”

New formulations for the hop-constrained minimum spanning tree problem via Sherali and Driscoll's tightened Miller–Tucker–Zemlin constraints

“…Other than the MTZ-based studies, there is a body of literature on flow-based and hop-indexed formulations for HMST, e.g., Gouveia [21,22], Gouveia and Requejo [23], Balakrishnan and Altinkemer [4], Pirkul and Soni [24], and Gouveia et al [5]. Even though the LP bounds are improved significantly over MTZ-based models in these studies, the formulations lead to very large LP models that may require huge computer storage requirements and excessive CPU times for high values of p and for intermediate or larger size networks.…”