Dvořák, P., Feldmann, A.E. orcid.org/0000-0001-6229-5332, Knop, D. orcid.org/0000-0003-2588-5709 et al. (3 more authors) (2021) Parameterized approximation schemes for Steiner Trees with small number of Steiner vertices. SIAM Journal on Discrete Mathematics, 35 (1). pp. 546-574. ISSN 0895-4801
Abstract
We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge- weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parameterization. In particular, on one hand Steiner Tree is known to be APX -hard, and W[2] -hard on the other, if parameterized by the number of nonterminals (Steiner vertices) in the optimum solution. In contrast to this, we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For none of these is an EPAS likely to exist for the studied parameter. For Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that approximating within any function of the studied parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree, but a PSAKS does not. We also prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021, Society for Industrial and Applied Mathematics. This is an author-produced version of a paper subsequently published in SIAM Journal on Discrete Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Steiner tree; Steiner forest; approximation algorithms; parameterized algorithms |
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: | 27 Jun 2023 14:57 |
Last Modified: | 27 Jun 2023 15:34 |
Status: | Published |
Publisher: | Society for Industrial & Applied Mathematics (SIAM) |
Refereed: | Yes |
Identification Number: | 10.1137/18m1209489 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:200945 |