Optimal discretization is fixed-parameter tractable

Given two disjoint sets W₁ and W₂ of points in the plane, the Optimal Discretization problem asks for the minimum size of a family of horizontal and vertical lines that separate W₁ from W₂, that is, in every region into which the lines partition the plane there are either only points of W₁, or only points of W₂, 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 W₁ × W₂ 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(^{k2 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.