Parameterized and approximation algorithms for coverings points with segments in the plane

We study parameterized and approximation algorithms for a variant of Set Cover, where the universe of elements to be covered consists of points in the plane and the sets with which the points should be covered are segments. We call this problem Segment Set Cover. We also consider a relaxation of the problem called $δ$-extension, where we need to cover the points by segments that are extended by a tiny fraction, but we compare the solution's quality to the optimum without extension. For the unparameterized variant, we prove that Segment Set Cover does not admit a PTAS unless $\mathsf{P}=\mathsf{NP}$, even if we restrict segments to be axis-parallel and allow $\frac{1}{2}$-extension. On the other hand, we show that parameterization helps for the tractability of Segment Set Cover: we give an FPT algorithm for unweighted Segment Set Cover parameterized by the solution size $k$, a parameterized approximation scheme for Weighted Segment Set Cover with $k$ being the parameter, and an FPT algorithm for Weighted Segment Set Cover with $δ$-extension parameterized by $k$ and $δ$. In the last two results, relaxing the problem is probably necessary: we prove that Weighted Segment Set Cover without any relaxation is $\mathsf{W}[1]$-hard and, assuming ETH, there does not exist an algorithm running in time $f(k)\cdot n^{o(k / \log k)}$. This holds even if one restricts attention to axis-parallel segments.