Given a geometric domain $P$, visibility-based search problems seek routes for one or more mobile agents (``watchmen'') to move within $P$ in order to be able to see a portion (or all) of $P$, while optimizing objectives, such as the length(s) of the route(s), the size (e.g., area or volume) of the portion seen, the probability of detecting a target distributed within $P$ according to a prior distribution, etc. The classic watchman route problem seeks a shortest route for an observer, with omnidirectional vision, to see all of $P$. In this paper we study bicriteria optimization problems for a single mobile agent within a polygonal domain $P$ in the plane, with the criteria of route length and area seen. Specifically, we address the problem of computing a minimum length route that sees at least a specified area of $P$ (minimum length, for a given area quota). We also study the problem of computing a length-constrained route that sees as much area as possible. We provide hardness results and approximation algorithms. In particular, for a simple polygon $P$ we provide the first fully polynomial-time approximation scheme for the problem of computing a shortest route seeing an area quota, as well as a (slightly more efficient) polynomial dual approximation. We also consider polygonal domains $P$ (with holes) and the special case of a planar domain consisting of a union of lines. Our results yield the first approximation algorithms for computing a time-optimal search route in $P$ to guarantee some specified probability of detection of a static target within $P$, randomly distributed in $P$ according to a given prior distribution.

翻译：给定几何区域$P$，基于可见性的搜索问题旨在为一个或多个移动智能体（“巡视员”）规划在$P$内的移动路径，使其能够观测$P$的部分（或全部）区域，同时优化目标函数，例如路径长度、观测区域大小（如面积或体积）、根据先验分布检测$P$内目标点的概率等。经典巡视员路径问题寻求一条最短路径，使具有全向视觉的观测者能够观测整个$P$。本文研究平面多边形区域$P$中单个移动智能体的双目标优化问题，以路径长度和观测面积为准则。具体而言，我们解决以下问题：计算一条观测至少指定面积$P$区域的最短路径（给定面积配额下的最短路径），并研究计算一条受长度约束、同时最大化观测面积的路径。我们给出了困难性结果和近似算法。特别地，对于简单多边形$P$，我们首次提出了针对计算满足面积配额的最短路径问题的完全多项式时间近似方案，以及一种（略高效）的多项式对偶近似算法。此外，我们还考虑了带孔多边形区域$P$，以及由直线并集构成的平面区域这一特例。我们的结果首次提供了近似算法，用于计算$P$中基于时间最优的搜索路径，从而保证以给定先验分布随机分布于$P$内的静态目标具有指定的检测概率。