2 research outputs found
Optimal Discretization is Fixed-parameter Tractable
Given two disjoint sets and of points in the plane, the Optimal
Discretization problem asks for the minimum size of a family of horizontal and
vertical lines that separate from , that is, in every region into
which the lines partition the plane there are either only points of , or
only points of , 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 -coordinates and the range of
-coordinates into as few segments as possible, maintaining that no pair of
points from 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 , where is the bound on the number of lines to find and 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.Comment: Accepted to ACM-SIAM Symposium on Discrete Algorithms (SODA 2021