Hitting Axis-Parallel Segments with Weighted Points

2026-05-14

AI Summary

This study addresses the geometric hitting set problem for axis-aligned line segments induced by a weighted point set in the plane, which is known to be NP-hard and APX-hard. The authors decompose the problem into horizontal and vertical subproblems and combine linear programming rounding with a systematic horizontal line rounding scheme and an exact repair strategy for the residual vertical subproblem. This approach breaks the longstanding factor-2 approximation barrier, achieving a (1 + 2/e)-approximation for the weighted case—the first such improvement. For the unweighted setting and the special case where one direction consists of full lines, the approximation ratios are further improved to (1 + 1/(e − 1)) and (1 + 1/e), respectively. The paper also presents efficient algorithms and a polynomial-time approximation scheme (PTAS) for natural extensions to d orientations and bounded-complexity subclasses.

Abstract

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.

Problem

• geometric hitting set
• axis-parallel segments
• weighted points
• approximation algorithm
• computational hardness

Innovation

• geometric hitting set
• LP rounding
• approximation algorithm
• axis-parallel segments
• PTAS