Covering a Set of Points with a Minimum Number of Lines

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 ∈ O(log 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 ∈ O( 4 √n). For the case when l ∈ Ω(log n) we suggest how to improve the time complexity of the exact algorithm by a factor exponential in l.