  Given two disjoint sets $W_1$ and $W_2$ of points in the plane, the Optimal
Discretization problem asks for the minimum size of a family of horizontal and
vertical lines that separate $W_1$ from $W_2$, that is, in every region into
which the lines partition the plane there are either only points of $W_1$, or
only points of $W_2$, or the region is empty. Equivalently, Optimal
Discretization can be phrased as a task of discretizing continuous variables:
we would like to discretize the range of $x$-coordinates and the range of
$y$-coordinates into as few segments as possible, maintaining that no pair of
points from $W_1 \times W_2$ are projected onto the same pair of segments under
this discretization.
  We provide a fixed-parameter algorithm for the problem, parameterized by the
number of lines in the solution. Our algorithm works in time $2^{O(k^2 \log k)}
n^{O(1)}$, where $k$ is the bound on the number of lines to find and $n$ is the
number of points in the input.
  Our result answers in positive a question of Bonnet, Giannopolous, and Lampis
[IPEC 2017] and of Froese (PhD thesis, 2018) and is in contrast with the known
intractability of two closely related generalizations: the Rectangle Stabbing
problem and the generalization in which the selected lines are not required to
be axis-parallel.
