Fine-grained complexity of coloring unit disks and balls
Résumé
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,
\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 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
\textsc{Independent Set} on unit disk graphs or for \textsc{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: \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.
Origine : Fichiers produits par l'(les) auteur(s)
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