Online Search for a Hyperplane in High-Dimensional Euclidean Space

We consider the online search problem in which a server starting at the origin of a $d$-dimensional Euclidean space has to find an arbitrary hyperplane. The best-possible competitive ratio and the length of the shortest curve from which each point on the $d$-dimensional unit sphere can be seen are within a constant factor of each other. We show that this length is in $\Omega(d)\cap O(d^{3/2})$

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP with lines does not admit a $(2-\epsilon)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log\log n)}$

On the exact complexity of hamiltonian cycle and q-colouring in disk graphs

We study the exact complexity of the Hamiltonian Cycle and the q-Colouring problem in disk graphs. We show that the Hamiltonian Cycle problem can be solved in (formula presented) on n-vertex disk graphs where the ratio of the largest and smallest disk radius is O(1). We also show that this is optimal: assuming the Exponential Time Hypothesis, there is no (formula presented)-time algorithm for Hamiltonian Cycle, even on unit disk graphs. We give analogous results for graph colouring: under the Expo-nential Time Hypothesis, for any fixed q, q-Colouring does not admit a (formula presented)-time algorithm, even when restricted to unit disk graphs, and it is solvable in (formula presented)-time on disk graphs

On geometric set cover for orthants

We study Set Cover for orthants: Given a set of points in a d-dimensional Euclidean space and a set of orthants of the form (−∞, p1] × . . . × (−∞, pd], select a minimum number of orthants so that every point is contained in at least one selected orthant. This problem draws its motivation from applications in multi-objective optimization problems. While for d = 2 the problem can be solved in polynomial time, for d > 2 no algorithm is known that avoids the enumeration of all size-k subsets of the input to test whether there is a set cover of size k. Our contribution is a precise understanding of the complexity of this problem in any dimension d ≥ 3, when k is considered a parameter: ⁃ For d = 3, we give an algorithm with runtime nO(√k), thus avoiding exhaustive enumeration. ⁃ For d = 3, we prove a tight lower bound of nΩ(√k) (assuming ETH). ⁃ For d ≥ 4, we prove a tight lower bound of nΩ(k) (assuming ETH). Here n is the size of the set of points plus the size of the set of orthants. The first statement comes as a corollary of a more general result: an algorithm for Set Cover for half-spaces in dimension 3. In particular, we show that given a set of points U in ℝ3, a set of half-spaces D in ℝ3, and an integer k, one can decide whether U can be covered by the union of at most k half-spaces from D in time |D|O(√k) · |U|O(1). We also study approximation for Set Cover for orthants. While in dimension 3 a PTAS can be inferred from existing results, we show that in dimension 4 and larger, there is no 1.05-approximation algorithm with runtime f(k) · no(k) for any computable f, where k is the optimum