  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 $\delta$-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 $\delta$-extension parameterized by $k$ and $\delta$. 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.
