k-Transmitter Watchman Routes

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