A parameterized algorithm for the hyperplane-cover problem

We consider the problem of covering a given set of points in the Euclidean space Rm by a small number k of hyperplanes of dimensions bounded by d, where d≤m. We present a very simple parameterized algorithm for the problem, and give thorough mathematical analysis to prove the correctness and derive the complexity of the algorithm. When the algorithm is applied on the standard hyperplane-cover problem in Rd, it runs in time O∗(k(d−1)k/1.3k), improving the previous best algorithm of running time O∗(kdk+d) for the problem. When the algorithm is applied on the line-cover problem in R2, it runs in time O∗(kk/1.35k), improving the previous best algorithm of running time O∗(k2k/4.84k) for the problem.