Chitnis, R., Feldmann, A.E. orcid.org/0000-0001-6229-5332 and Manurangsi, P. (2018) Parameterized approximation algorithms for bidirected steiner network problems. In: Azar, Y., Bast, H. and Herman, G., (eds.) Leibniz International Proceedings in Informatics, LIPIcs. 26th Annual European Symposium on Algorithms (ESA 2018), 20-24 Aug 2018, Helsinki, Finland. Leibniz International Proceedings in Informatics, 112 . Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik , Dagstuhl, Germany ISBN 978-3-95977-081-1
Abstract
The DIRECTED STEINER NETWORK (DSN) problem takes as input a directed edge-weighted graph G = (V, E) and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N ⊆ G for which there is an s → t path for each (s,t) ∈ D. It is known that this problem is notoriously hard as there is no k1/4-o(1)-approximation algorithm under Gap-ETH, even when parameterizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k. For the BI-DSNPLANAR problem, the aim is to compute a planar optimum solution N ⊆ G in a bidirected graph G, i.e. for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of BI-DSNPLANAR, unless FPT=W[1]. Additionally we study several generalizations of BI-DSNPLANAR and obtain upper and lower bounds on obtainable runtimes parameterized by k. One important special case of DSN is the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem, for which the solution network N ⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists when parameterized by k [Chitnis et al., IPEC 2013]. We show a tight inapproximability result: under Gap-ETH there is no (2 - ϵ)-approximation algorithm parameterized by k (for any ϵ > 0). To the best of our knowledge, this is the first example of a W[1]-hard problem admitting a non-trivial parameterized approximation factor which is also known to be tight! Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Editors: |
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Copyright, Publisher and Additional Information: | © 2018 The Authors. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (https://creativecommons.org/licenses/by/3.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
Keywords: | Directed Steiner Network; Strongly Connected Steiner Subgraph; Parameterized Approximations; Bidirected Graphs; Planar Graphs |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 28 Jun 2023 14:28 |
Last Modified: | 28 Jun 2023 14:28 |
Status: | Published |
Publisher: | Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik |
Series Name: | Leibniz International Proceedings in Informatics |
Refereed: | Yes |
Identification Number: | 10.4230/LIPIcs.ESA.2018.20 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:200959 |