The Parameterized Complexity of Some Geometric Problems in Unbounded Dimension

We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension d: i) Given n points in ? d , compute their minimum enclosing cylinder. ii) Given two n-point sets in ? d , 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 ?(d)-time lower bound (under the Exponential Time Hypothesis).

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