Subexponential Algorithms for Clique Cover on Unit Disk and Unit Ball Graphs

Abstract: In Clique Cover, given a graph $G$ and an integer $k$, the task is to partition the vertices of $G$ into $k$ cliques. Clique Cover on unit ball graphs has a natural interpretation as a clustering problem, where the objective function is the maximum diameter of a cluster. Many classical NP-hard problems are known to admit $2^{{O(n^{(1} - 1/d)})}$-time algorithms on unit ball graphs in $\mathbb{R}<sup>d$ [de Berg et al., SIAM J. Comp 2018]. A notable exception is the Maximum Clique problem, which admits a polynomial-time algorithm on unit disk graphs and a subexponential algorithm on unit ball graphs in $\mathbb{R}<sup>3$, but no subexponential algorithm on unit ball graphs in dimensions 4 or larger, assuming the ETH [Bonamy et al., JACM 2021]. In this work, we show that Clique Cover also suffers from a "curse of dimensionality", albeit in a significantly different way compared to Maximum Clique. We present a $2<sup>{O(\sqrt{n})}$-time algorithm for unit disk graphs and argue that it is tight under the ETH. On the other hand, we show that Clique Cover does not admit a $2<sup>{o(n)}$-time algorithm on unit ball graphs in dimension $5$, unless the ETH fails.