Hitting Axis-Parallel Segments with Weighted Points

We study a geometric hitting-set problem in which the input consists of a set $P$ of weighted points and a family $S=H\cup V$ of axis-parallel segments in the plane. The goal is to select a minimum-weight subset of $P$ that hits every segment in $S$. Even restricted geometric hitting-set problems are known to be computationally hard, and for axis-parallel segments the standard decomposition into horizontal and vertical sub-instances yields only a simple factor-$2$ approximation. We present an LP-rounding algorithm that breaks the factor-2 barrier. For the weighted problem, we obtain a randomized $(1+2/e)$-approximation by combining systematic rounding on horizontal lines with an exact repair step on residual vertical sub-instances. In the unweighted case, a sharper analysis gives a $(1+1/(e-1))$-approximation. Finally, we consider the case where one of the sub-instances consists of lines instead of line segments, a problem considered by Fekete et al. (Geometric Hitting Set for Segments of Few Orientations, Theor. Comp. Sys., 62 (2) 2018),. In this case, we improve their result to obtain an approximation factor of $1+1/e$ and show that the problem is APX-hard. We also present algorithms for the generalization to $d$ orientations, as well as PTASes for bounded-complexity subclasses of the unweighted Hitting Set problem.