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

B.M.P. Jansen, M. Pilipczuk, M. 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 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 poly(k)?
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.
Original languageEnglish
Article number1707.01797
Pages (from-to)1-22
JournalarXiv
Publication statusPublished - 2017

Keywords

  • Data Structures and Algorithms
  • Computational Complexity

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