Tight Kernels for Covering with Points and Polynomials

Description

The Point Hyperplane Cover problem in $R d$ takes as input a set of $n$ points in $R d$ 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 Cover problem in $R d$ takes as input a family $F$ of D-degree polynomials from a vector space $R$ in $R d$ , and determines whether there is a set of at most $k$ points in $R d$ that hit all the polynomials in $F$. Here, a point p is said to hit a polynomial $f$ if $f (p) = 0$. For both problems, we exhibit tight kernels where $k$ is the parameter. We also exhibit a tight kernel for the Projective Point Hyperplane Cover problem, where the hyperplanes that are allowed to cover the points must all contain a fixed point, and the fixed point cannot be included in the solution set of points.

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