Tight Kernels for Covering and Hitting: Point Hyperplane Cover and Polynomial Point Hitting Set

Abstract

The Point Hyperplane Cover problem in $\mathbb {R}^d$ takes as input a set of n points in $\mathbb {R}^d$ and a positive integer k. The objective is to cover all the given points with a set of at most k hyperplanes. The D-Polynomial Points Hitting Set (D-Polynomial Points HS) problem in $\mathbb {R}^d$ takes as input a family $\mathcal {F}$ of D-degree polynomials from a vector space $\mathcal {R}$ in $\mathbb {R}^d$, and determines whether there is a set of at most k points in $\mathbb {R}^d$ that hit all the polynomials in $\mathcal {F}$. For both problems, we exhibit tight kernels where k is the parameter.