Geometric Hitting Set for Segments of Few Orientations

TL;DR

This paper investigates geometric hitting set problems for segments with few orientations, providing approximation algorithms, hardness results, and polynomial solutions for specific cases relevant to path monitoring.

Contribution

It introduces new approximation algorithms and hardness results for geometric hitting set problems with limited segment orientations.

Findings

Polynomial solvability for vertical lines and horizontal rays.

Approximation algorithms for segments with 3 slopes.

Hardness and approximation bounds for various segment configurations.

Abstract

We study several natural instances of the geometric hitting set problem for input consisting of sets of line segments (and rays, lines) having a small number of distinct slopes. These problems model path monitoring (e.g., on road networks) using the fewest sensors (the "hitting points"). We give approximation algorithms for cases including (i) lines of 3 slopes in the plane, (ii) vertical lines and horizontal segments, (iii) pairs of horizontal/vertical segments. We give hardness and hardness of approximation results for these problems. We prove that the hitting set problem for vertical lines and horizontal rays is polynomially solvable.