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The Steiner tree problem with hop constraints

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  • S. Voß

Abstract

The Steiner tree problem in graphs is to determine a minimum cost subgraph of a givengraph spanning a set of specified vertices. In certain telecommunication networks, additionalconstraints such as, e.g., reliability constraints, have to be observed. Assume that a certainreliability is associated with each arc of the network, measuring the probability that therespective arc is operational. In case there has to be a guarantee that each message sent froma root vertex to a specified vertex reaches its destination with a certain probability, so‐calledhop constraints may be used to model the respective generalization. In this paper, we discussthe Steiner tree problem with hop constraints, i.e., a generalization of Steiner's problem ingraphs where the number of arcs (hops) between a root node and any of the specified verticesis limited. A mathematical programming formulation is provided and extended to handleproblem instances of moderate size. As the Steiner tree problem with hop constraints is NP‐hard,a simple heuristic is developed and an exchange procedure based on the tabu searchmetastrategy is applied to improve given solutions. Numerical results are discussed for anumber of problem instances derived from, e.g., well‐known benchmark instances of Steiner'sproblem in graphs. Copyright Kluwer Academic Publishers 1999

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  • S. Voß, 1999. "The Steiner tree problem with hop constraints," Annals of Operations Research, Springer, vol. 86(0), pages 321-345, January.
    • Handle: RePEc:spr:annopr:v:86:y:1999:i:0:p:321-345:10.1023/a:1018967121276
    DOI: 10.1023/A:1018967121276
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    Cited by:

    1. Konak, Abdullah, 2012. "Network design problem with relays: A genetic algorithm with a path-based crossover and a set covering formulation," European Journal of Operational Research, Elsevier, vol. 218(3), pages 829-837.
    2. Nigam, Ashutosh & Agarwal, Yogesh K., 2014. "Optimal relay node placement in delay constrained wireless sensor network design," European Journal of Operational Research, Elsevier, vol. 233(1), pages 220-233.
    3. Kang, Jangha & Park, Kyungchul & Park, Sungsoo, 2009. "Optimal multicast route packing," European Journal of Operational Research, Elsevier, vol. 196(1), pages 351-359, July.
    4. Fu, Zhang-Hua & Hao, Jin-Kao, 2014. "Breakout local search for the Steiner tree problem with revenue, budget and hop constraints," European Journal of Operational Research, Elsevier, vol. 232(1), pages 209-220.
    5. Gouveia, Luis & Requejo, Cristina, 2001. "A new Lagrangean relaxation approach for the hop-constrained minimum spanning tree problem," European Journal of Operational Research, Elsevier, vol. 132(3), pages 539-552, August.
    6. Li, Xiangyong & Aneja, Y.P. & Huo, Jiazhen, 2012. "Using branch-and-price approach to solve the directed network design problem with relays," Omega, Elsevier, vol. 40(5), pages 672-679.
    7. Gouveia, Luis & Leitner, Markus & Ljubić, Ivana, 2014. "Hop constrained Steiner trees with multiple root nodes," European Journal of Operational Research, Elsevier, vol. 236(1), pages 100-112.
    8. Cabral, Edgar Alberto & Erkut, Erhan & Laporte, Gilbert & Patterson, Raymond A., 2007. "The network design problem with relays," European Journal of Operational Research, Elsevier, vol. 180(2), pages 834-844, July.

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