On Formulations and Methods for the Hop-Constrained Minimum Spanning Tree Problem

Dahl, Geir; Gouveia, Luís; Requejo, Cristina

doi:10.1007/978-0-387-30165-5_19

Cited by 48 publications

(61 citation statements)

References 25 publications

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“…Computational tests are performed using specially-structured test problems from the literature (e.g., Dahl et al, 2006). Test problems consist of three 20-node, three 40-node, and three 60-node, complete networks with 210, 820, and 1830 edges, respectively.…”

Section: Test Problemsmentioning

confidence: 99%

“…An alternative formulation is offered by Gouveia (1996) based on directed or undirected multicommodity flows (MCF). Even though the LP bounds of MCF formulations are considerably better than those of MTZ formulations, MCF formulations lead to very large integer programming models whose LP relaxations require excessive solution times and core storage as the network size and p get larger (Dahl et al, 2006). Additional improvements are obtained for small values of p using a hop-indexed formulation (Gouveia, 1998).…”

Section: Introductionmentioning

confidence: 99%

“…MTZ formulations are much more compact in the numbers of variables and constraints than MCF based formulations. There are problem instances, for example, where an MTZ formulation finds an integer feasible solution in seconds whereas an MCF formulation cannot even solve the LP relaxation in days (Dahl et al, 2006). Computational results also indicate that it may take a considerable amount of time to arrive at an optimal integer solution even if the gap between the LP and integer optimal values is quite small.…”

Section: Introductionmentioning

confidence: 99%

“…Even though this approach gives better bounds than the aforementioned ones, it is dependent on the value of p and performs poorly as p increases. Dahl et al (2004) introduce a new formulation for the HMST problem using only natural design variables and an exponential number of constraints composed of the so-called jump inequalities that are shown to be facet-defining. Their proposed formulation uses fewer variables but has weaker LP bounds than MCF formulations.…”

Section: Introductionmentioning

confidence: 99%

“…Computational results indicate that the LP bounds are better as p increases than those reported in previous studies. Dahl et al (2006) summarize the aforementioned approaches. Kerivin and Mahjoub (2005) give a survey of several network design problems with hop constraints and methods to solve them.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller–Tucker–Zemlin constraints

Akgün

Tansel

2011

European Journal of Operational Research

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a b s t r a c tGiven an undirected network with positive edge costs and a natural number p, the Hop-Constrained Minimum Spanning Tree problem (HMST) is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops (edges). In this paper, we develop new formulations for HMST. The formulations are based on Miller-Tucker-Zemlin (MTZ) subtour elimination constraints, MTZ-based liftings in the literature offered for HMST, and a new set of topologyenforcing constraints. We also compare the proposed models with the MTZ-based models in the literature with respect to linear programming relaxation bounds and solution times. The results indicate that the new models give considerably better bounds and solution times than their counterparts in the literature and that the new set of constraints is competitive with liftings to MTZ constraints, some of which are based on well-known, strong liftings of Desrochers and Laporte (1991).

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Section: Test Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller–Tucker–Zemlin constraints

Akgün

Tansel

2011

European Journal of Operational Research

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Models and branch‐and‐cut algorithms for the Steiner tree problem with revenues, budget and hop constraints

Costa¹,

Cordeau²,

Laporte³

2008

Networks

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The Steiner tree problem with revenues, budget and hop constraints is a variant of the Steiner tree problem with two main modifications: (a) besides the costs associated with arcs, there are also revenues associated with the vertices, and (b) there are additional budget and hop constraints, which impose limits on the total cost of the network and on the number of edges between any vertex and the root, respectively. This article introduces and compares several mathematical models for this problem and describes two branch-and-cut algorithms, which solve to optimality instances with up to 500 vertices and 625 edges.

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Reload cost trees and network design

2011

Self Cite

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In this article, we consider the notion of "reload costs" in network design. Reload costs occur naturally in many different settings including telecommunication networks using diverse technologies. However, reload costs have not been studied extensively in the literature. Given that reload costs occur naturally in many settings, we are motivated by the desire to develop "good" models for network design problems involving reload costs. In this article, and as a first step in this direction, we propose and discuss the reload cost spanning tree problem (RCSTP). We show that the RCSTP is NP-complete. We discuss several ways of modeling network design problems with reload costs. These involve models that expand the original graph significantly-to a directed line graph and a colored graph-to model reload costs. We show that the different modeling approaches lead to models with the same linear programming bound. We then discuss several variations of reload cost spanning tree and network design problems, and discuss both their complexity and models for these variations. To assess the effectiveness of the proposed models to solve RCSTP instances, we present results taken from instances with up to 50 nodes, 300 edges, and nine technologies for several variations of the problem.

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On Formulations and Methods for the Hop-Constrained Minimum Spanning Tree Problem

Cited by 48 publications

References 25 publications

New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller–Tucker–Zemlin constraints

New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller–Tucker–Zemlin constraints

Models and branch‐and‐cut algorithms for the Steiner tree problem with revenues, budget and hop constraints

Reload cost trees and network design

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