Feldmann, A.E. orcid.org/0000-0001-6229-5332 and Rai, A. (2021) On extended formulations for parameterized Steiner trees. In: Golovach, P. A. and Zehavi, M., (eds.) Leibniz International Proceedings in Informatics, LIPIcs. 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), 08-10 Sep 2021, Virtual event. Leibniz International Proceedings in Informatics, 214 . Schloss Dagstuhl - Leibniz-Zentrum , Dagstuhl, Germany , 18:1-18:16. ISBN 978-3-95977-216-7
Abstract
We present a novel linear program (LP) for the Steiner Tree problem, where a set of terminal vertices needs to be connected by a minimum weight tree in a graph G = (V, E) with non-negative edge weights. This well-studied problem is NP-hard and therefore does not have a compact extended formulation (describing the convex hull of all Steiner trees) of polynomial size, unless P=NP. On the other hand, Steiner Tree is fixed-parameter tractable (FPT) when parameterized by the number k of terminals, and can be solved in O(3k|V | + 2k|V |2) time via the Dreyfus-Wagner algorithm. A natural question thus is whether the Steiner Tree problem admits an extended formulation of comparable size. We first answer this in the negative by proving a lower bound on the extension complexity of the Steiner Tree polytope, which, for some constant c > 0, implies that no extended formulation of size f(k)2cn exists for any function f. However, we are able to circumvent this lower bound due to the fact that the edge weights are non-negative: we prove that Steiner Tree admits an integral LP with O(3k|E|) variables and constraints. The size of our LP matches the runtime of the Dreyfus-Wagner algorithm, and our poof gives a polyhedral perspective on this classic algorithm. Our proof is simple, and additionally improves on a previous result by Siebert et al. [2018], who gave an integral LP of size O((2k/e)k)|V |O(1)
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: | © 2021 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
Keywords: | Steiner trees; integral linear program; extension complexity; fixed-parameter tractability |
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 11:33 |
Last Modified: | 28 Jun 2023 12:00 |
Status: | Published |
Publisher: | Schloss Dagstuhl - Leibniz-Zentrum |
Series Name: | Leibniz International Proceedings in Informatics |
Refereed: | Yes |
Identification Number: | 10.4230/LIPIcs.IPEC.2021.18 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:200941 |