Approximation Algorithms for k-Line Center

Given a set P of n points in ℝd and an integer k ≥ 1, let w* denote the minimum value so that P can be covered by k cylinders of radius at most w*. We describe an algorithm that, given P and an ɛ > 0, computes k cylinders of radius at most (1 + ɛ)w* that cover P. The running time of the algorithm is O(n log n), with the constant of proportionality depending on k, d, and ɛ. We first show that there exists a small “certificate” Q ⊆ P, whose size does not depend on n, such that for any k-cylinders that cover Q, an expansion of these cylinders by a factor of (1+ɛ) covers P. We then use a well-known scheme based on sampling and iterated re-weighting for computing the cylinders.