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 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).

• Type: Conference
• Year: 2009
• Where: IWPEC
• Authors: Panos Giannopoulos, Christian Knauer, Günter Rote
• Keywords: Applied Computing, Dimension D, Fundamental Geometric Problems, IWPEC 2009, Minimum Enclosing Cylinder