Hyperbolic intersection graphs and (quasi)-polynomial time

Abstract: We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in $d$-dimensional hyperbolic space, which we denote by $\mathbb{H}<sup>d$. Using a new separator theorem, we show that unit ball graphs in $\mathbb{H}<sup>d$ enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in $2<sup>{O(n<sup>{1-1/(d-1)})}$ time for any fixed $d\geq 3$, while the same problems need $2<sup>{O(n<sup>{1-1/d})}$ time in $\mathbb{R}<sup>d$. We also show that these algorithms in $\mathbb{H}<sup>d$ are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in $\mathbb{H}<sup>2$, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial ($n^{O(\log n)}$) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and $3$-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in $\mathbb{H}<sup>2$ have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require $2<sup>{\Omega(\sqrt{n})}$ time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching $n^{\Omega(\log n)}$ lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.