Applications of incidence bounds in point covering problems

Abstract

In the Line Cover problem a set of n points is given and the task is to cover the points using either the minimum number of lines or at most k lines. In Curve Cover, a generalization of Line Cover, the task is to cover the points using curves with d degrees of freedom. Another generalization is the Hyperplane Cover problem where points in d-dimensional space are to be covered by hyperplanes. All these problems have kernels of polynomial size, where the parameter is the minimum number of lines, curves, or hyperplanes needed.First we give a non-parameterized algorithm for both problems in O * (2 n ) (where the O * (•) notation hides polynomial factors of n) time and polynomial space, beating a previous exponential-space result. Combining this with incidence bounds similar to the famous Szemerédi-Trotter bound, we present a Curve Cover algorithm with running time O * (Ck/ log k) (d−1)k , where C is some constant.Our result improves the previous best times O * (k/1.35) k for Line Cover (where d = 2), O * k dk for general Curve Cover, as well as a few other bounds for covering points by parabolas or conics. We also present an algorithm for Hyperplane Cover in R 3 with running time O * (Ck 2 / log 1/5 k) k , improving on the previous time of O * (k 2 /1.3) k .

Nondegenerate spheres in four dimensions

“…See also Remark 1.2ii. iv The second term mn 2/3 is the same with the second term in Elekes-Tóth's bound (1) for hyperplanes in R 4 . This is consistent with Apfelbaum-Sharir's result (3).…”

“…Under this more general framework, we extend Apfelbaum-Sharir's method in [7] to some other families of surfaces in R 3 and use that to prove a new upper bound for incidences with nondegenerate spheres in four dimensions. 1 What they actually proved is that the maximum number of β-nondegenerate, k-rich (i.e. containing at…”

“…These incidence bounds with nondegenerate hyperplanes and spheres have many applications in discrete geometry. For example, the bound (1) was used in [1] to derive an efficient algorithm for the point covering problem in R 3 . It was also used in [6] to derive a lower bound on the maximum size of a complete bipartite subgraph one can find in a point-hyperplane incidence graph with many edges.…”

Extending Erdős–Beck's theorem to higher dimensions

“…Point covering problem is a famous problem in computation geometry which asks for efficient ways to cover n points in space using lines, curves or hyperplanes, hypersurfaces. In [1], the authors use the point-hyperplane bound in theorem 1.10 to derive a good hyperplane covering algorithm. However, it only works in R 3 because of the strong result when d = 3.…”