The parameterized complexity of some geometric problems in unbounded dimension

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 nΩ(d)-time lower bound (under the Exponential Time Hypothesis). © 2009 Springer-Verlag.