Tight Kernels for Covering with Points and Polynomials
Résumé
The Point Hyperplane Cover problem in Rd takes as input a set of n points in Rd 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 Rd takes as input a family F of D-degree polynomials from a vector space R in Rd , and determines whether there is a set of at most k points in Rd 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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