Turing kernelization for finding long paths in graph classes excluding a topological minor

Bart M.P. Jansen, Marcin Pilipczuk, Marcin Wrochna

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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 boundedsize 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 kO(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 K3,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.

Original languageEnglish
Title of host publication12th International Symposium on Parameterized and Exact Computation, IPEC 2017
EditorsD. Lokshtanov, N. Nishimura
Place of PublicationDagstuhl
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Number of pages13
ISBN (Electronic)978-3-95977-051-4
DOIs
Publication statusPublished - 1 Feb 2018
Event12th International Symposium on Parameterized and Exact Computation (IPEC 2017) - Vienna, Austria
Duration: 6 Sep 20178 Sep 2017
Conference number: 12
https://algo2017.ac.tuwien.ac.at/ipec

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
Volume89

Conference

Conference12th International Symposium on Parameterized and Exact Computation (IPEC 2017)
Abbreviated titleIPEC 2017
CountryAustria
CityVienna
Period6/09/178/09/17
Internet address

Keywords

  • Excluded topological minor
  • K-path
  • Long path
  • Modulator
  • Turing kernel

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