On the Parameterized Complexity of Red-Blue Points Separation

Authors Édouard Bonnet, Panos Giannopoulos, Michael Lampis

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Édouard Bonnet
Panos Giannopoulos
Michael Lampis

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Édouard Bonnet, Panos Giannopoulos, and Michael Lampis. On the Parameterized Complexity of Red-Blue Points Separation. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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).
  • red-blue points separation
  • geometric problem
  • W[1]-hardness
  • FPT algorithm
  • ETH-based lower bound


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