On extended formulations for parameterized steiner trees
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10437397" target="_blank" >RIV/00216208:11320/21:10437397 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.4230/LIPIcs.IPEC.2021.18" target="_blank" >https://doi.org/10.4230/LIPIcs.IPEC.2021.18</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.4230/LIPIcs.IPEC.2021.18" target="_blank" >10.4230/LIPIcs.IPEC.2021.18</a>
Alternative languages
Result language
angličtina
Original language name
On extended formulations for parameterized steiner trees
Original language description
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) (C) Andreas Emil Feldmann and Ashutosh Rai; licensed under Creative Commons License CC-BY 4.0
Czech name
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Czech description
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Classification
Type
D - Article in proceedings
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
<a href="/en/project/GX19-27871X" target="_blank" >GX19-27871X: Efficient approximation algorithms and circuit complexity</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Article name in the collection
16th International Symposium on Parameterized and Exact Computation (IPEC 2021)
ISBN
978-3-95977-216-7
ISSN
1868-8969
e-ISSN
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Number of pages
16
Pages from-to
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Publisher name
Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Place of publication
Dagstuhl
Event location
Online
Event date
Sep 8, 2021
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
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