The two-level diameter constrained spanning tree problem

Abstract

In this article, we introduce the two-level diameter constrained spanning tree problem (2-DMSTP), which generalizes the classical DMSTP by considering two sets of nodes with different latency requirements. We first observe that any feasible solution to the 2-DMSTP can be viewed as a DMST that contains a diameter constrained Steiner tree. This observation allows us to prove graph theoretical properties related to the centers of each tree which are then exploited to develop mixed integer programming formulations, valid inequalities, and symmetry breaking constraints. In particular, we propose a novel modeling approach based on a three-dimensional layered graph. In an extensive computational study we show that a branch-and-cut algorithm based on the latter model is highly effective in practice.

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Correspondence to Markus Leitner.

Additional information

L. Gouveia is supported by National Funding from FCT—Fundação para a Ciência e Tecnologia, under the project: PEst-OE/MAT/UI0152. M. Leitner is supported by the Austrian Science Fund (FWF) under grant I892-N23. I. Ljubić is supported by the APART Fellowship of the Austrian Academy of Sciences. These supports are greatly acknowledged.

Appendix

Appendix

See Tables 5, 6, 7 and 8.

Table 5 Results for solving the LP relaxations on instances with 31 and 41 nodes: numbers of instances solved within the given time limit (#\(_\mathrm{solved}\)), geometric means of CPU-times in seconds, and average LP gaps (%) with respect to instance graph and \(|P|\)
Table 6 Numbers of instances solved to optimality (#\(_\mathrm{solved}\)), geometric means of CPU-times in seconds, and average optimality gaps in (%) with respect to instance graph and \(|P|\)
Table 7 Numbers of instances solved to optimality (\(\#_\mathrm{solved}\)), geometric means of CPU-times in seconds, and average optimality gaps in (%) with respect to \(D\) and \(D'\) for Euclidean instances
Table 8 Numbers of instances solved to optimality (\(\#_\mathrm{solved}\)), geometric means of CPU-times in seconds, and average optimality gaps in (%) with respect to \(D\) and \(D'\) for random instances

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Gouveia, L., Leitner, M. & Ljubić, I. The two-level diameter constrained spanning tree problem. Math. Program. 150, 49–78 (2015). https://doi.org/10.1007/s10107-013-0743-z

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Keywords

  • Networks/graphs: tree algorithms
  • Integer programming: formulations
  • Layered graphs

Mathematics Subject Classification (2000)

  • 90C11
  • 90C27
  • 90C57