On the Parameterized Complexity of Red-Blue Points Separation

Abstract

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/log 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).

Locations

• Middlesex University Research Repository (Middlesex University Of London) - View - PDF
• arXiv (Cornell University) - View - PDF
• Base Institutionnelle de Recherche de l'université Paris-Dauphine (BIRD) (University Paris-Dauphine) - View - PDF
• City Research Online (City University London) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
• Leibniz-Zentrum für Informatik (Schloss Dagstuhl) - View - PDF