On the approximability of covering points by lines and related problems

Given a set P of n points in the plane, sc Covering Points by Lines is the problem of finding a minimum-cardinality set L of lines such that every point p ∈ P is incident to some line ℓ ∈ L. As a geometric variant of sc Set Cover, sc Covering Points by Lines is still NP-hard. Moreover, it has been proved to be APX-hard, and hence does not admit any polynomial-time approximation scheme unless P = NP@. In contrast to the small constant approximation lower bound implied by APX-hardness, the current best approximation ratio for sc Covering Points by Lines is still O(log n), namely the ratio achieved by the greedy algorithm for sc Set Cover. In this paper, we give a lower bound of Omega(log n) on the approximation ratio of the greedy algorithm for sc Covering Points by Lines. We also study several related problems including sc Maximum Point Coverage by Lines, sc Minimum-Link Covering Tour, sc Minimum-Link Spanning Tour, and sc Min-Max-Turn Hamiltonian Tour. We show that all these problems are either APX-hard or at least NP-hard. In particular, our proof of APX-hardness of sc Min-Max-Turn Hamiltonian Tour sheds light on the difficulty of sc Bounded-Turn-Minimum-Length Hamiltonian Tour, a problem proposed by Aggarwal et al. at SODA 1997.