2013/12/09 by Dumitrescu, Adrian, Jiang, Minghui
#Computational Geometry (cs.CG) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.1312.2549
Given a set P of n points in the plane, \sc Covering Points by Lines is the problem of finding a minimum-cardinality set Ł of lines such that every point p ∈ P is incident to some line ℓ ∈ Ł. 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 Ω(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.