Multi-level steiner trees

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

Research output: Contribution to journalArticle

Abstract

In the classical Steiner tree problem, given an undirected, connected graphG = (V, E) with non-negative edge costs and a set of terminalsT ° V, the objective is to find aminimum-cost tree 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 article,we study a natural generalization, the multi-level Steiner tree (MLST) problem: Given a nested sequence of terminals T1 ° V, compute nested trees E ° E E E ° E1 ° E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under various names, including Multi-level Network Design, Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-tier tree. Several approximation results are known. We first present two simple O(ℓ)-approximation heuristics. Based on these, we introduce a rudimentary composite algorithm that generalizes the above heuristics, and determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio using at most 2ℓ Steiner tree computations.We compare these heuristics experimentally on various instances of up to 500 vertices using three different network generation models. We also present several integer linear programming formulations for the MLST problem and compare their running times on these instances. To our knowledge, the composite algorithm achieves the best approximation ratio for up to ℓ = 100 levels,which is sufficient for most applications, such as network visualization or designing multi-level infrastructure.

Original languageEnglish (US)
Article number3368621
JournalACM Journal of Experimental Algorithmics
Volume24
Issue number1
DOIs
StatePublished - Dec 2019

    Fingerprint

Keywords

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

ASJC Scopus subject areas

  • Theoretical Computer Science

Cite this

Ahmed, R., Angelini, P., Sahneh, F. D., Efrat, A., Glickenstein, D., Gronemann, M., Heinsohn, N., Kobourov, S. G., Spence, R., Watkins, J., & Wolff, A. (2019). Multi-level steiner trees. ACM Journal of Experimental Algorithmics, 24(1), [3368621]. https://doi.org/10.1145/3368621