License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2021.18
URN: urn:nbn:de:0030-drops-154010
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/15401/
Feldmann, Andreas Emil ;
Rai, Ashutosh
On Extended Formulations For Parameterized Steiner Trees
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(3^k|V|+2^k|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)2^{cn} 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(3^k|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)}.
BibTeX - Entry
@InProceedings{feldmann_et_al:LIPIcs.IPEC.2021.18,
author = {Feldmann, Andreas Emil and Rai, Ashutosh},
title = {{On Extended Formulations For Parameterized Steiner Trees}},
booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
pages = {18:1--18:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-216-7},
ISSN = {1868-8969},
year = {2021},
volume = {214},
editor = {Golovach, Petr A. and Zehavi, Meirav},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/15401},
URN = {urn:nbn:de:0030-drops-154010},
doi = {10.4230/LIPIcs.IPEC.2021.18},
annote = {Keywords: Steiner trees, integral linear program, extension complexity, fixed-parameter tractability}
}
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Keywords: |
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Steiner trees, integral linear program, extension complexity, fixed-parameter tractability |
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Collection: |
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16th International Symposium on Parameterized and Exact Computation (IPEC 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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22.11.2021 |
