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Turing Kernelization for Finding Long Paths in Graphs Excluding a Topological Minor

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

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  • Turing kernel
  • long path
  • k-path
  • excluded topological minor
  • modulator

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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 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.

Cite As Get BibTex

Bart M. P. Jansen, Marcin Pilipczuk, and Marcin Wrochna. Turing Kernelization for Finding Long Paths in Graphs Excluding a Topological Minor. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.IPEC.2017.23

Author Details

Bart M. P. Jansen
Marcin Pilipczuk
Marcin Wrochna

References

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