Covering a set of points with a minimum number of lines

Abstract

We consider the minimum line covering problem: given a set S of n points in the plane, we want to find the smallest number l of straight lines needed to cover all n points in S. We show that this problem can be solved in O(n log l) time if l ε O(log 1-ε n), and that this is optimal in the algebraic computation tree model (we show that the Ω(n log l) lower bound holds for all values of l up to O(√n)). Furthermore, a O(log l)-factor approximation can be found within the same O(n log l) time bound if l ε O(4√n). For the case when l ε Ω(log n) we suggest how to improve the time complexity of the exact algorithm by a factor exponential in l. © Springer-Verlag Berlin Heidelberg 2006.