Hyperbolic intersection graphs and (quasi)-polynomial time

We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in d-dimensional hyperbolic space, which we denote by H-d. Using a new separator theorem, we show that unit ball graphs in Hd 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(O(n1-1/(d-1)) time for any fixed d >= 3, while the same problems need 2(O(n1-1/(d)) time in R-d. We also show that these algorithms in H-d are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in H-2, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasipolynomial (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 H-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(Omega(root 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. 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.