Observation Routes and External Watchman Routes

TL;DR

The Observation Route Problem is introduced: given a set of pairwise disjoint compact regions in the plane, find a shortest tour such that an observer walking along this tour can see some point in each region from some point of the tour.

Abstract

We introduce the Observation Route Problem ($\textsf{ORP}$) defined as follows: Given a set of $n$ pairwise disjoint compact regions in the plane, find a shortest tour (route) such that an observer walking along this tour can see (observe) some point in each region from some point of the tour. The observer does \emph{not} need to see the entire boundary of an object. The tour is \emph{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 ($\textsf{TSPN}$) and the External Watchman Route Problem ($\textsf{EWRP}$). 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 $\textsf{P} = \textsf{NP}$ 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 $\textsf{NP}$-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, 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 $n \geq 5$, there exists a convex polygon with $n$ sides and all angles obtuse such that its perimeter is \emph{not} a shortest external watchman route. This refutes a conjecture by Absar and Whitesides (2006).

Figures (11)

• Figure 1: An observation route (the blue point) and an external watchman route (dotted, in red) for a set of five triangles.
• Figure 2: Left: three convex polygons in a bounding box. Right: the visibility region $V$ of the triangle is a polygon with two holes.
• Figure 3: Left: This family can be only observed from single points far away up or down. Right: This family can be observed from any single point on a horizontal line that separates the upper chain of squares from the lower one.
• Figure 4: Local replacements to obtain $T'$ from $T$.
• Figure 5: Two translates of a convex polygon in a cone.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 1
• proof
• Lemma 2
• proof