Algorithms for Halfplane Coverage and Related Problems

Given in the plane a set of points and a set of halfplanes, we consider the problem of computing a smallest subset of halfplanes whose union covers all points. In this paper, we present an $O(n^{{4/3}\log{5/3}n\log{O(1)}\log} n)$-time algorithm for the problem, where $n$ is the total number of all points and halfplanes. This improves the previously best algorithm of $n^{{10/3}2{O(\log*n)}$} time by roughly a quadratic factor. For the special case where all halfplanes are lower ones, our algorithm runs in $O(n\log n)$ time, which improves the previously best algorithm of $n^{{4/3}2{O(\log*n)}$} time and matches an $\Omega(n\log n)$ lower bound. Further, our techniques can be extended to solve a star-shaped polygon coverage problem in $O(n\log n)$ time, which in turn leads to an $O(n\log n)$-time algorithm for computing an instance-optimal $\epsilon$-kernel of a set of $n$ points in the plane. Agarwal and Har-Peled presented an $O(nk\log n)$-time algorithm for this problem in SoCG 2023, where $k$ is the size of the $\epsilon$-kernel; they also raised an open question whether the problem can be solved in $O(n\log n)$ time. Our result thus answers the open question affirmatively.