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Hitting a Set of Line Segments with One or Two Discrete Centers

Given the scheduling model of bike-sharing, we consider the problem of hitting a set of n axis-parallel line segments in R by a square (and two squares) whose center(s) must lie on some line segment(s) such that the (maximum) edge length of the square(s) is minimized. Under a different model, we also consider the cases when one needs to compute one (and two) centers on some edge(s) of a tree of size m, where n labeled segments must be hit, and the objective is to minimize the maximum path length from the labeled segments to the nearer center(s). We give three linear-time algorithms and an O(n log n) algorithm for the four problems in consideration.