The parameterized complexity of some geometric problems in unbounded dimension

TL;DR

This paper investigates the computational complexity of several fundamental geometric problems in high-dimensional spaces, establishing their W[1]-hardness and exponential time lower bounds when parameterized by dimension.

Contribution

It proves that key geometric problems are W[1]-hard in high dimensions and provides exponential lower bounds under the ETH, highlighting their computational difficulty.

Findings

All problems are W[1]-hard when parameterized by dimension.

No fixed-parameter tractable algorithms are likely to exist for these problems.

There are exponential time lower bounds under ETH for these problems.

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 $d$. %and hence not solvable in ${O}(f(d)n^c)$ time, for any computable function $f$ and constant $c$ %(unless FPT=W[1]). Our reductions also give a $n^{\Omega(d)}$-time lower bound (under the Exponential Time Hypothesis).