# On the Parameterized Complexity of Red-Blue Points Separation

We study the following geometric separation problem: Given a set R of red points and a set B of blue points in the plane, find a minimum-size set of lines that separate R from B. We show that, in its full generality, parameterized by the number of lines k in the solution, the problem is unlikely to be solvable significantly faster than the brute-force n^O(k)-time algorithm, where n is the total number of points. Indeed, we show that an algorithm running in time f(k)n^o(k/ k), for any computable function f, would disprove ETH. Our reduction crucially relies on selecting lines from a set with a large number of different slopes (i.e., this number is not a function of k). Conjecturing that the problem variant where the lines are required to be axis-parallel is FPT in the number of lines, we show the following preliminary result. Separating R from B with a minimum-size set of axis-parallel lines is FPT in the size of either set, and can be solved in time O^*(9^|B|) (assuming that B is the smallest set).

## Authors

• 27 publications
• 4 publications
• 22 publications
• ### Red-Blue Point Separation for Points on a Circle

Given a set R of red points and a set B of blue points in the plane, the...
05/12/2020 ∙ by Neeldhara Misra, et al. ∙ 0

• ### Optimal Discretization is Fixed-parameter Tractable

Given two disjoint sets W_1 and W_2 of points in the plane, the Optimal ...
03/05/2020 ∙ by Stefan Kratsch, et al. ∙ 0

• ### Exact Method for Generating Strategy-Solvable Sudoku Clues

A Sudoku puzzle often has a regular pattern in the arrangement of initia...
05/28/2020 ∙ by Kohei Nishikawa, et al. ∙ 0

• ### Searching for Point Locations using Lines

Versions of the following problem appear in several topics such as Gamma...
06/18/2021 ∙ by Michelle Cordier, et al. ∙ 0

• ### Separating Geometric Data with Minimum Cost: Two Disjoint Convex Hulls

In this study, a geometric version of an NP-hard problem ("Almost 2-SAT"...
06/18/2021 ∙ by Bahram Sadeghi Bigham, et al. ∙ 0

• ### Quantum algorithms for computational geometry problems

We study quantum algorithms for problems in computational geometry, such...
04/19/2020 ∙ by Andris Ambainis, et al. ∙ 0