Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs
Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Meirav Zehavi · Discrete & Computational Geometry · 2019
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 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.