On the Computational Complexity of some problems from Combinatorial Geometry

We study several canonical decision problems that arise from the most famous theorems from combinatorial geometry. We show that these are W[1]-hard (and NP-hard) if the dimension is part of the input by fpt-reductions (which are actually ptime-reductions) from the d-Sum problem. Among others, we show that computing the minimum size of a Caratheodory set and a Helly set and certain decision versions of the Ham-Sandwich cut problem are W[1]-hard. Our reductions also imply that the problems we consider cannot be solved in time n (where n is the size of the input), unless the Exponential-Time Hypothesis (ETH) is false. Our technique of embedding d-Sum into a geometric setting is conceptually much simpler than direct fpt-reductions from purely combinatorial W[1]-hard problems (like the clique problem) and has great potential to show (parameterized) hardness and (conditional) lower bounds for many other problems.