Abstract
We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension d:
i) Given n points in ℝd, compute their minimum enclosing cylinder.
ii) Given two n-point sets in ℝd, decide whether they can be separated by two hyperplanes.
iii) Given a system of n linear inequalities with d variables, find a maximum size feasible subsystem.
We show that (the decision versions of) all these problems are W[1]-hard when parameterized by the dimension d. Our reductions also give a n Ω(d)-time lower bound (under the Exponential Time Hypothesis).
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Giannopoulos, P., Knauer, C., Rote, G. (2009). The Parameterized Complexity of Some Geometric Problems in Unbounded Dimension. In: Chen, J., Fomin, F.V. (eds) Parameterized and Exact Computation. IWPEC 2009. Lecture Notes in Computer Science, vol 5917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11269-0_16
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DOI: https://doi.org/10.1007/978-3-642-11269-0_16
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