License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2019.25
URN: urn:nbn:de:0030-drops-102642
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10264/
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Eiben, Eduard ; Knop, Dusan ; Panolan, Fahad ; Suchý, Ondrej

Complexity of the Steiner Network Problem with Respect to the Number of Terminals

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Abstract

In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals T subseteq V(G) with |T|=q, and an (unweighted) directed request graph R with V(R)=T. Our task is to output a subgraph H subseteq G of the minimum cost such that there is a directed path from s to t in H for all st in A(R).
It is known that the problem can be solved in time |V(G)|^{O(|A(R)|)} [Feldman and Ruhl, SIAM J. Comput. 2006] and cannot be solved in time |V(G)|^{o(|A(R)|)} even if G is planar, unless the Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, the reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time |V(G)|^{o(q)}, unless ETH fails. Therefore, there is a significant gap in the complexity with respect to q in the exponent.
We show that Directed Steiner Network is solvable in time f(q)* |V(G)|^{O(c_g * q)}, where c_g is a constant depending solely on the genus of G and f is a computable function. We complement this result by showing that there is no f(q)* |V(G)|^{o(q^2/ log q)} algorithm for any function f for the problem on general graphs, unless ETH fails.

BibTeX - Entry

@InProceedings{eiben_et_al:LIPIcs:2019:10264,
  author =	{Eduard Eiben and Dusan Knop and Fahad Panolan and Ondrej Such{\'y}},
  title =	{{Complexity of the Steiner Network Problem with Respect to the Number of Terminals}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Rolf Niedermeier and Christophe Paul},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10264},
  doi =		{10.4230/LIPIcs.STACS.2019.25},
  annote =	{Keywords: Directed Steiner Network, Planar Graphs, Parameterized Algorithms, Bounded Genus, Exponential Time Hypothesis}
}

Keywords: Directed Steiner Network, Planar Graphs, Parameterized Algorithms, Bounded Genus, Exponential Time Hypothesis
Collection: 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)
Issue Date: 2019
Date of publication: 12.03.2019

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