  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.
