Observation routes and external watchman routes

We introduce the Observation Route Problem (ORP) defined as follows: Given a set of n pairwise disjoint obstacles (regions) in the plane, find a shortest tour (route) such that an observer walking along this tour can see (observe) each obstacle from some point of the tour. The observer does not need to see the entire boundary of an obstacle. The tour is not allowed to intersect the interior of any region (i.e., the regions are obstacles and therefore out of bounds). The problem exhibits similarity to both the Traveling Salesman Problem with Neighborhoods (TSPN) and the External Watchman Route Problem (EWRP). We distinguish two variants: the range of visibility is either limited to a bounding rectangle, or unlimited. We obtain the following results:

(I) Given a family of n disjoint convex bodies in the plane, computing a shortest observation route does not admit a $(c\log n)$-approximation unless $P=\mathrm{NP}$ for an absolute constant $c>0$. (This holds for both limited and unlimited vision.)

(II) Given a family of disjoint convex bodies in the plane, computing a shortest external watchman route is $\mathrm{NP}$-hard. (This holds for both limited and unlimited vision; and even for families of axis-aligned squares.)

(III) Given a family of n disjoint fat convex polygons in the plane, an observation tour whose length is at most $O(\log n)$ times the optimal can be computed in polynomial time. (This holds for limited vision.)

(IV) For every $n\ge 5$, there exists a convex polygon with n sides and all angles obtuse such that its perimeter is not a shortest external watchman route. This refutes a conjecture by Absar and Whitesides (2006).