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

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... table 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). 1 Introduction We study the parameterized complexity of the following Red-Blue Separation problem: Given a set R of red points and a set B of blue points in the plane and a positive integer k, find a set of at most k lines that together separate R from B (or report that such a set does not exist). Separation here means that each cell in the arrangement induced by the lines in the solution is either monochromatic, i.e., contains points of one color only, or empty. Equivalently, R is separated from B if every straight-line segment with one endpoint in R and the other one in B is intersected by at least one line in the solution. Note here that we opt for strict separation that is, no point in R ∪ B is on a separating line. Let n := |R ∪ B|.