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Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs

  • Fedor V. Fomin
  • Daniel Lokshtanov
  • Fahad PanolanEmail author
  • Saket Saurabh
  • Meirav Zehavi
Article
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Abstract

We give algorithms with running time \(2^{\mathcal {O}({\sqrt{k}\log {k}})} \cdot n^{\mathcal {O}(1)}\) for the following problems. Given an n-vertex unit disk graph G and an integer k, decide whether G contains
  • a path on exactly/at least k vertices,

  • a cycle on exactly k vertices,

  • a cycle on at least k vertices,

  • a feedback vertex set of size at most k, and

  • a set of k pairwise vertex-disjoint cycles.

For the first three problems, no subexponential time parameterized algorithms were previously known. For the remaining two problems, our algorithms significantly outperform the previously best known parameterized algorithms that run in time \(2^{\mathcal {O}(k^{0.75}\log {k})} \cdot n^{\mathcal {O}(1)}\). Our algorithms are based on a new kind of tree decompositions of unit disk graphs where the separators can have size up to \(k^{\mathcal {O}(1)}\) and there exists a solution that crosses every separator at most \(\mathcal {O}(\sqrt{k})\) times. The running times of our algorithms are optimal up to the \(\log {k}\) factor in the exponent, assuming the exponential time hypothesis.

Keywords

Longest path Longest cycle Cycle packing Feedback vertex set Unit disk graph Unit square graph Parameterized complexity 

Mathematics Subject Classification

68W01 68W40 68Q25 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.University of CaliforniaSanta BarbaraUSA
  3. 3.The Institute of Mathematical SciencesHBNIChennaiIndia
  4. 4.Ben-Gurion University of the NegevBeer-ShevaIsrael

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