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.

翻译：我们研究了集合覆盖问题（Set Cover）的一种变体的参数化与近似算法。在该变体中，待覆盖的基元素为平面上的点，而用于覆盖点的集合均为线段。我们将此问题称为线段集合覆盖（Segment Set Cover）。同时，我们还考虑了该问题的一种松弛形式，即$\delta$-扩展：需用扩展微小比例的线段覆盖点，但将解的质量与无扩展时的最优解进行比较。针对非参数化变体，我们证明：即使限制线段与坐标轴平行并允许$\frac{1}{2}$-扩展，线段集合覆盖问题仍不存在多项式时间近似方案（PTAS），除非$\mathsf{P}=\mathsf{NP}$。另一方面，我们表明参数化有助于提升线段集合覆盖问题的可解性：我们给出了无权线段集合覆盖问题在解规模$k$参数化下的固定参数可解（FPT）算法；针对加权线段集合覆盖问题，以$k$为参数给出了参数化近似方案；针对带$\delta$-扩展的加权线段集合覆盖问题，给出了以$k$和$\delta$为参数的FPT算法。后两个结果中，松弛问题可能是必要的：我们证明，无任何松弛的加权线段集合覆盖问题是$\mathsf{W}[1]$-困难的，且假设指数时间假设（ETH）成立，不存在运行时间为$f(k)\cdot n^{o(k / \log k)}$的算法。即使仅限于坐标轴平行线段，该结论仍成立。