The limited blessing of low dimensionality

Abstract

We are studying d-dimensional geometric problems that have algorithms with 1−1/d appearing in the exponent of the running time, for example, in the form of 2 n 1−1/d or n k 1−1/d . This means that these algorithms perform somewhat better in low dimensions, but the running time is almost the same for all large values d of the dimension. Our main result is showing that for some of these problems the dependence on 1 − 1/d is best possible under a standard complexity assumption. We show that, assuming the Exponential Time Hypothesis,• d-dimensional Euclidean TSP on n points cannot be solved in time 2 O(n 1−1/d− ) for any > 0, and• the problem of finding a set of k pairwise nonintersecting d-dimensional unit balls/axis parallel unit cubes cannot be solved in time f (k)n o(k 1−1/d ) for any computable function f .These lower bounds essentially match the known algorithms for these problems. To obtain these results, we first prove lower bounds on the complexity of Constraint Satisfaction Problems (CSPs) whose constraint graphs are d-dimensional grids. We state the complexity results on CSPs in a way to make them convenient starting points for problem-specific reductions to particular d-dimensional geometric problems and to be reusable in the future for further results of similar flavor.Theory 1 This variant of TSP is also known as path-TSP. Our lower bound also holds for tour-TSP, i.e. the variant where one seeks to find a cycle visiting all vertices. In order to simplify the discussion, we restrict our attention to path-TSP for now; on Section 4 we explain how our proof can be modified to obtain the same lower bound for cycle-TSP.

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).…”

“…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 ;…”

Fractal Dimension and Lower Bounds for Geometric Problems

“…In this problem given a set of n unit balls in R d , we seek to find a set of k pairwise non-intersecting balls. It is known that this problem can be solved in time n O(k 1−1/d ) , for any d ≥ 2 [1,19], and that there is no algorithm with running time f (k)n o(k 1−1/d ) , for any computable function f , assuming ETH [19] (see also [17]). The upper bound has been generalized for fractal dimension as follows: It has been shown that when the set of centers of the balls has fractal dimension δ, the problem can be solved in time n O(k 1−1/δ log n) [23].…”

“…Informally, a typical NP-hardness reduction for some geometric problem in the plane works as follows: One encodes some known computationally hard problem by constructing "gadgets" that are arranged in a grid-like fashion in R 2 (see, e.g. [19]). More generally, for problems in R d , the gadgets are arranged along some d-dimensional grid.…”

The Homogeneous Broadcast Problem in Narrow and Wide Strips II: Lower Bounds

“…exact computation. Grid Tiling has also been extended to higher dimensions [15], and successfully used to establish lower bounds for coloring unit disk and unit ball graphs [16]. Most recently, it has been used to study the parameterized complexity of Steiner Tree in planar graphs [17,18].…”