  We consider the watchman route problem for a $k$-transmitter watchman:
standing at point $p$ in a polygon $P$, the watchman can see $q\in P$ if
$\overline{pq}$ intersects $P$'s boundary at most $k$ times -- $q$ is
$k$-visible to $p$. Traveling along the $k$-transmitter watchman route, either
all points in $P$ or a discrete set of points $S\subset P$ must be $k$-visible
to the watchman. We aim for minimizing the length of the $k$-transmitter
watchman route.
  We show that even in simple polygons the shortest $k$-transmitter watchman
route problem for a discrete set of points $S\subset P$ is NP-complete and
cannot be approximated to within a logarithmic factor (unless P=NP), both with
and without a given starting point. Moreover, we present a polylogarithmic
approximation for the $k$-transmitter watchman route problem for a given
starting point and $S\subset P$ with approximation ratio $O(\log^2(|S|\cdot n)
\log\log (|S|\cdot n) \log(|S|+1))$ (with $|P|=n$).
