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).

翻译：我们引入如下定义的观测路线问题（$\textsf{ORP}$）：给定平面上一组$n$个两两不相交的紧致区域，寻找一条最短的巡游路线（路径），使得沿此路线行走的观测者能够从路线上的某个点看到（观察到）每个区域中的某个点。观测者无需看到物体的完整边界。该路线不允许与任何区域的内部相交（即区域被视为障碍物并因此禁止通行）。该问题与邻域旅行商问题（$\textsf{TSPN}$）及外部巡逻者路线问题（$\textsf{EWRP}$）均具有相似性。我们区分两种变体：可见范围要么局限于一个边界矩形，要么无限制。我们得到以下结果：（I）给定平面上一族$n$个不相交凸体，计算最短观测路线不存在$(c\log n)$-近似算法，除非对于绝对常数$c>0$有$\textsf{P} = \textsf{NP}$（该结论对有限视野和无限视野均成立）。（II）给定平面上一族不相交凸体，计算最短外部巡逻者路线是$\textsf{NP}$-困难的（该结论对有限视野和无限视野均成立，甚至对于轴对齐正方形族也是如此）。（III）给定一族$n$个不相交的胖凸多边形，可以在多项式时间内计算出一条长度不超过最优解$O(\log{n})$倍的观测路线（该结论对有限视野成立）。（IV）对于每个$n \geq 5$，存在一个所有角均为钝角的$n$边凸多边形，其周长并非最短外部巡逻者路线。这反驳了Absar和Whitesides（2006）的一个猜想。