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 $|P|=n$).

翻译：我们研究面向$k$-透射监视员的监视员路径问题：对于多边形$P$中的点$p$，若线段$\overline{pq}$与$P$的边界相交次数不超过$k$次（即$q$相对于$p$是$k$-可见的），则监视员在$p$处可看见点$q\in P$。沿$k$-透射监视员路径行进时，要求多边形内所有点或离散点集$S\subset P$对监视员保持$k$-可见。我们的目标是极小化$k$-透射监视员路径的长度。研究表明，即使在简单多边形中，面向离散点集$S\subset P$的最短$k$-透射监视员路径问题在给定起点和未给定起点两种情形下均为NP-完全问题，且除非P=NP，否则不存在对数因子以内的近似算法。此外，我们针对给定起点及$S\subset P$的情形，给出了近似比为$O(\log^2(|S|\cdot n) \log\log (|S|\cdot n) \log(|S|+1))$（其中$|P|=n$）的多对数近似算法。