eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-03-02
23:1
23:13
10.4230/LIPIcs.IPEC.2017.23
article
Turing Kernelization for Finding Long Paths in Graphs Excluding a Topological Minor
Jansen, Bart M. P.
Pilipczuk, Marcin
Wrochna, Marcin
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol089-ipec2017/LIPIcs.IPEC.2017.23/LIPIcs.IPEC.2017.23.pdf
Turing kernel
long path
k-path
excluded topological minor
modulator