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 ∈ O(log^1 − ε n), and that this is optimal in the algebraic computation tree model (we show that the Ω(nlog l) lower bound holds for all values of l up to $O(\sqrt n)$). Furthermore, a O(log l)-factor approximation can be found within the same O(n log l) time bound if $l \in O(\sqrt[4]{n})$. 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.