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On the Parameterized Complexity of Red-Blue Points Separation

Giannopoulos, P. ORCID: 0000-0002-6261-1961, Bonnet, E. and Lampis, M. (2019). On the Parameterized Complexity of Red-Blue Points Separation. Journal of Computational Geometry, 10(1), pp. 181-206. doi: 10.20382/jocg.v10i1a7

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 bruteforce nO(k) -time algorithm, where n is the total number of points. Indeed, we show that an algorithm running in time f(k)nᵒ(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 smaller set).

Publication Type: Article
Subjects: G Geography. Anthropology. Recreation > GA Mathematical geography. Cartography
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Departments: School of Mathematics, Computer Science & Engineering > Computer Science > Computing
URI: http://openaccess.city.ac.uk/id/eprint/22279
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