eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-03-02
8:1
8:13
10.4230/LIPIcs.IPEC.2017.8
article
On the Parameterized Complexity of Red-Blue Points Separation
Bonnet, Édouard
Giannopoulos, Panos
Lampis, Michael
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).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol089-ipec2017/LIPIcs.IPEC.2017.8/LIPIcs.IPEC.2017.8.pdf
red-blue points separation
geometric problem
W[1]-hardness
FPT algorithm
ETH-based lower bound