Multi-level steiner trees

Reyan Ahmed, Patrizio Angelini, Faryad Darabi Sahneh, Alon Efrat, David A Glickenstein, Martin Gronemann, Niklas Heinsohn, Stephen G Kobourov, Richard Spence, Joseph C Watkins, Alexander Wol

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

In the classical Steiner tree problem, one is given an undirected, connected graph G = (V, E) with non-negative edge costs and a set of terminals T ⊆ V . The objective is to find a minimum-cost edge set E ⊆ E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of ρ = ln(4) + ε < 1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals T 1 ⊂ · · · ⊂ T k ⊆ V , compute nested edge sets E 1 ⊆ · · · ⊆ E k ⊆ E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under names such as Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two natural heuristics with approximation factor O(k). Based on these, we introduce a composite algorithm that requires 2 k Steiner tree computations. We determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio and needs at most 2k Steiner tree computations. We compare five algorithms experimentally on several classes of graphs using four types of graph generators. We also implemented an integer linear program for MLST to provide ground truth.

Our combined algorithm outperforms the others both in theory and in practice when the number of levels is small (k ≤ 22), which works well for applications such as designing multi-level infrastructure or network visualization.

Original languageEnglish (US)
Title of host publication17th Symposium on Experimental Algorithms, SEA 2018
EditorsGianlorenzo D'Angelo
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770705
DOIs
Publication statusPublished - Jun 1 2018
Event17th Symposium on Experimental Algorithms, SEA 2018 - L'Aquila, Italy
Duration: Jun 27 2018Jun 29 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume103
ISSN (Print)1868-8969

Conference

Conference17th Symposium on Experimental Algorithms, SEA 2018
CountryItaly
CityL'Aquila
Period6/27/186/29/18

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Keywords

  • Approximation algorithm
  • Multi-level graph representation
  • Steiner tree

ASJC Scopus subject areas

  • Software

Cite this

Ahmed, R., Angelini, P., Sahneh, F. D., Efrat, A., Glickenstein, D. A., Gronemann, M., ... Wol, A. (2018). Multi-level steiner trees. In G. D'Angelo (Ed.), 17th Symposium on Experimental Algorithms, SEA 2018 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 103). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SEA.2018.15