Fine-grained complexity of coloring unit disks and balls

• Published in: Journal of computational geometry
• Main Authors: Csaba, Biro, Bonnet, Edouard, Marx, Dániel, Tillmann, Miltzow, Rzążewski, Paweł
• Format: Journal Article
• Language: English
• Published: Carleton University, Computational Geometry Laboratory 2018
• Subjects: Computational Complexity; Computational Geometry; Computer Science; Data Structures and Algorithms
• ISSN: 1920-180X; 1920-180X

On planar graphs, many classic algorithmic problems enjoy a certain ``square rootphenomenon'' and can be solved significantly fasterthan what is known to be possible on general graphs: for example,\textsc{Independent Set}, \textsc{3-Coloring}, \textsc{Hamiltonian Cycle}, \textsc{Dominating Set} can be solved in time$2^{O(\sqrt{n})}$ on an $n$-vertex planar graph, while no $2^{o(n)}$algorithms exist for general graphs, assuming the Exponential TimeHypothesis (ETH). The square root in the exponent seems to be bestpossible for planar graphs: assuming the ETH, the running time for theseproblems cannot be improved to $2^{o(\sqrt{n})}$. In some cases, asimilar speedup can be obtained for 2-dimensional geometric problems,for example, there are $2^{O(\sqrt{n}\log n)}$ time algorithms for\textsc{Independent Set} on unit disk graphs or for \textsc{TSP} on2-dimensional point sets.In this paper, we explore whether such a speedup is possible for geometric coloring problems. On the one hand, geometric objects can behave similarly to planar graphs: \textsc{3-Coloring} can be solved in time $2^{O(\sqrt{n})}$ on the intersection graph of $n$ disks in the plane and, assuming the ETH, there is no such algorithm with running time $2^{o(\sqrt{n})}$. On the other hand, if the number $\ell$ of colors is part of the input, then no such speedup is possible: Coloring the intersection graph of $n$ unit disks with $\ell$ colors cannot be solved in time $2^{o(n)}$, assuming the ETH. More precisely, we exhibit a smooth increase of complexity as the number $\ell$ of colors increases: If we restrict the number of colors to $\ell=\Theta(n^{\alpha})$ for some $0\le \alpha\le 1$, then the problem of coloring the intersection graph of $n$ disks with $\ell$ colors\begin{itemize}\item can be solved in time $\exp \left( O(n^{\frac{1+\alpha}{2}}\log n) \right)=\exp \left( O(\sqrt{n\ell}\log n) \right)$, and%using a combination of fairly standard techniques, and\item cannot be solved in time $\exp \left ( o(n^{\frac{1+\alpha}{2}})\right )=\exp \left( o(\sqrt{n\ell}) \right)$, even on unit disks, unless the ETH fails.\end{itemize}More generally, we consider the problem of coloring $d$-dimensional balls in the Euclidean space and obtain analogous results showing that the problem \begin{itemize}\item can be solved in time $\exp \left( O(n^{\frac{d-1+\alpha}{d}}\log n) \right)$ $=\exp \left( O(n^{1-1/d}\ell^{1/d}\log n) \right)$, and\item cannot be solved in time $\exp \left(O(n^{\frac{d-1+\alpha}{d}-\epsilon})\right)= \exp \left(O(n^{1-1/d-\epsilon}\ell^{1/d})\right)$ for any $\epsilon>0$, even for unit balls, unless the ETH fails.\end{itemize}Finally, we prove that fatness is crucial to obtain subexponential algorithms for coloring. We show that existence of an algorithm coloring an intersection graph of segments using a constant number of colors in time $2^{o(n)}$ already refutes the ETH.