A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs

Abstract

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.

Hyperbolic intersection graphs and (quasi)-polynomial time

“…From the perspective of treewidth and exact algorithms, unit disk graphs have some intriguing properties: they are potentially dense, (may have treewidth Ω(n)), but they still exhibit the "square root phenomenon" for several problems just as planar and minor-free graphs do; so for example one can solve I S or 3 in these classes in 2 O( √ n) time [29], while these problems would require 2 Θ(n) time in general graphs unless the Exponential Time Hypothesis (ETH) [23] fails. In R d , the best I S running time for unit ball graphs is 2 Θ(n 1−1/d ) [31,14]. Note that d-dimensional Euclidean space has bounded doubling dimension, or in other words, Euclidean space has polynomial growth: balls of radius r have volume poly(r).…”

“…Naturally, getting a sublinear bound on separators or treewidth itself is not possible, since cliques are unit ball graphs. Therefore, we use the partition and weighting scheme developed by De Berg et al [14]. Given an intersection graph G = (V, E), one de nes a partition P of the vertex set V ; initially, it is useful to think of a partition into cliques using a tiling of the underlying space where each tile has a small diameter.…”

“…(We give more thorough de nitions in Section 3.) For example, unit ball graphs in R d have a clique partition P such that their Pattened treewidth is O(n 1−1/d ) [14].…”

“…There have been several papers studying I S , D S , H C , q C , etc. in unit ball graphs in Euclidean space [24,25,31,4,18,14], all concluding that 2 O(n 1−1/d ) is the optimal running time for these problems in R d . In this paper, we show that a similar phenomenon occurs in H d , but shifted by one dimension: the problems can be solved in 2 O( √ n) time in H 3 , just as in R 2 ;…”

“…On the other hand, this embedding does not facilitate getting algorithms in H d , and it is also not useful for establishing algorithms and lower bounds in H 2 . In H d , we give a new separator theorem, which can be regarded as the hyperbolic analogue of various separator theorems known for (similarly sized) fat objects in Euclidean space [34,42,1,14]. 1 In H 2 , we use a novel approach to gain tight treewidth bounds: our argument builds on the isoperimetric inequality [43] and on the treewidth bound of k-outerplanar graphs [8].…”