  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 $p \in P$ is incident to some line $\ell \in \L$. 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 $=$<EMAIL_ADDRESS>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.
