Fine-Grained Complexity of Coloring Unit Disks and Balls

On planar graphs, many classic algorithmic problems enjoy a certain ``square root phenomenon'' and can be solved significantly faster than what is known to be possible on general graphs: for example, Independent Set , 3-Coloring , Hamiltonian Cycle , 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 Time Hypothesis (ETH). The square root in the exponent seems to be best possible for planar graphs: assuming the ETH, the running time for these problems cannot be improved to $2^{o(\sqrt{n})}$. In some cases, a similar speedup can be obtained for 2-dimensional geometric problems, for example, there are $2^{O(\sqrt{n}\log n)}$ time algorithms for Independent Set on unit disk graphs or for TSP on 2-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: 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 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 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. More generally, we consider the problem of coloring $d$-dimensional balls in the Euclidean space and obtain analogous results showing that the problem 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 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. 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.