Hyperbolic intersection graphs and (quasi)-polynomial time
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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 H^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^O(n^1-1/(d-1)) time for any fixed d≥ 3, while the same problems need 2^O(n^1-1/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 quasi-polynomial (n^O( 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 ply, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require 2^Ω(√(n)) time under ETH in constant-ply Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching n^Ω( n) lower bound under ETH. As far as we know, this is the first natural problem with a quasi-polynomial lower bound that is shown to be tight.
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