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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2017.23
URN: urn:nbn:de:0030-drops-85576
URL: https://drops.dagstuhl.de/opus/volltexte/2018/8557/
Jansen, Bart M. P. ;
Pilipczuk, Marcin ;
Wrochna, Marcin
Turing Kernelization for Finding Long Paths in Graphs Excluding a Topological Minor
Abstract
The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k-PATH admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size k^{O(1)}?
We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and K_{3,t}-minor-free graphs. Moreover, we show that k-PATH even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path has a separation that can safely be reduced after communication with the oracle.
BibTeX - Entry
@InProceedings{jansen_et_al:LIPIcs:2018:8557,
author = {Bart M. P. Jansen and Marcin Pilipczuk and Marcin Wrochna},
title = {{Turing Kernelization for Finding Long Paths in Graphs Excluding a Topological Minor}},
booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
pages = {23:1--23:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-051-4},
ISSN = {1868-8969},
year = {2018},
volume = {89},
editor = {Daniel Lokshtanov and Naomi Nishimura},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8557},
URN = {urn:nbn:de:0030-drops-85576},
doi = {10.4230/LIPIcs.IPEC.2017.23},
annote = {Keywords: Turing kernel, long path, k-path, excluded topological minor, modulator}
}
Keywords: |
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Turing kernel, long path, k-path, excluded topological minor, modulator |
Collection: |
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12th International Symposium on Parameterized and Exact Computation (IPEC 2017) |
Issue Date: |
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2018 |
Date of publication: |
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02.03.2018 |