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

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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.

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Notes

  1. The paper [23] does not consider unit square graphs, but the arguments it presents for unit disk graphs can be adapted to handle unit square graphs as well.

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Correspondence to Fahad Panolan.

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The preliminary version of the paper appeared in the proceedings of ICALP 2017. The research leading to the results of the paper is supported by Pareto-Optimal Parameterized Algorithms, ERC Starting Grant 715744, Parameterized Approximation, ERC Starting Grant 306992, Rigorous Theory of Preprocessing, ERC Advanced Investigator Grant 267959, and NFR MULTIVAL project.

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Fomin, F.V., Lokshtanov, D., Panolan, F. et al. Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs. Discrete Comput Geom 62, 879–911 (2019). https://doi.org/10.1007/s00454-018-00054-x

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