Separating points by axis-parallel lines

We study the problem of separating n points in the plane, no two of whi h have the same xor yoordinate, using a minimum number of verti al and horizontal lines avoiding the points, so that ea h ell of the subdivision ontains at most one point. Extending previous NP-hardness results due to Freimer et al. we prove that this problem and some variants of it are APX-hard. We give a 2-approximation algorithm for this problem, and a d-approximation algorithm for the d-dimensional variant, in whi h the points are to be separated using axis-parallel hyperplanes. To this end, we redu e the point separation problem to the re tangle stabbing problem studied by Gaur et al. Their approximation algorithm uses LP-rounding. We present an alternative LP-rounding pro edure whi h also works for the re tangle stabbing problem. We show that the integrality ratio of the LP is exa tly 2.