We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to a wide range of geometric intersection graphs (intersections of similarly sized fat objects), yielding algorithms with running time 2 O(n 1−1/d ) for any fixed dimension d ≥ 2 for many well known graph problems, including Independent Set, r-Dominating Set for constant r, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms work on the graph itself, i.e., do not require any geometric information. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques.The lower bound framework is based on a constructive embedding of graphs into ddimensional grids, and it allows us to derive matching 2 Ω(n 1−1/d ) lower bounds under the Exponential Time Hypothesis even in the much more restricted class of d-dimensional induced grid graphs.

We give an algorithmic and lower bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to intersection graphs of similarly-sized fat objects, yielding algorithms with running time 2 O(n 1 - 1/d ) for any fixed dimension d \geq 2 for many well-known graph problems, including Independent Set, r-Dominating Set for constant r, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms are representationagnostic, i.e., they work on the graph itself and do not require the geometric representation. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques. The lower bound framework is based on a constructive embedding of graphs into d-dimensional grids, and it allows us to derive matching 2 \Omega (n 1 - 1/d ) lower bounds under the exponential time hypothesis even in the much more restricted class of d-dimensional induced grid graphs.

We study exact algorithms for EUCLIDEAN TSP in R d . In the early 1990s algorithms with n O( √ n)running time were presented for the planar case, and some years later an algorithm with n O(n 1−1/d ) running time was presented for any d 2. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on EUCLIDEAN TSP, except for a lower bound stating that the problem admits no 2 O(n 1−1/d−ε ) algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of EUCLIDEAN TSP by giving a 2 O(n 1−1/d ) algorithm and by showing that a 2 o(n 1−1/d ) algorithm does not exist unless ETH fails. affirmatively by Arora [1] who provided a PTAS with running time n(log n) O( √ d/ε) d−1 . Independently, Mitchell [18] designed a PTAS in R 2 . The running time was improved to 2 (1/ε) O(d) n + (1/ε) O(d) n log n by Rao and Smith [22]. Hence, the computational complexity of the approximation problem has essentially been settled. Results on exact algorithms for EUCLIDEAN TSP-these are the topic of our paper-are also quite different from those on the general problem. The best known algorithm for the general case runs, as already remarked, in exponential time, and there is no 2 o(n) algorithm under ETH due to classical reductions for HAMILTONIAN CYCLE [5, Theorem 14.6]. EUCLIDEAN TSP, on the other hand, is solvable in subexponential time. For the planar case this has been shown in the early 1990s by Kann [14] and independently by Hwang, Chang and Lee [12], who presented an algorithm with an n O( √ n) running time. Both algorithms use a divide-and-conquer approach that relies on finding a suitable separator. The approach taken by Hwang, Chang and Lee is based on considering a triangulation of the point set such that all segments of the tour appear in the triangulation, and then observing that the resulting planar graph has a separator of size O( √ n). Such a separator can be guessed in n O( √ n) ways, leading to a recursive algorithm with n O( √ n) running time. It seems hard to extend this approach to higher dimensions. Kann obtains his separator in a more geometric way, using the fact that in an optimal tour, there cannot be too many long edges that are relatively close together-see the Packing Property we formulate in Section 2. This makes it possible to compute a separator that is crossed by O( √ n) edges of an optimal tour, which can be guessed in n O( √ n)

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 I S , D S , S T , and H C can be solved in 2 O(n 1−1/(d−1) ) time for any xed 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(log n) ) algorithms for all of the studied problems, while in the case of H C and 3 C 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 Ω( √ n) time under ETH in constant maximum degree Euclidean unit disk graphs.Finally, we complement our quasi-polynomial algorithm for I S in noisy uniform disk graphs with a matching n Ω(log n) lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.

The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity.Our main technical contribution is a novel balanced separator theorem for planar δ-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed δ ⩾ 0, we can find balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph.An important advantage of our separator is that the union of our separator (vertex set Z) with any subset of the connected components of G − Z induces again a planar δ-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that size of separator is poly(δ) • log n.As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar δ-hyperbolic graphs. We prove that Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant δ, running inWe also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no n o(δ) -time algorithm on planar δ-hyperbolic graphs, unless ETH fails.