Generalized Watchman Route Problem with Discrete View Cost

The watchman route problem (WRP) refers to planning a closed curve, called a watchman route, in a polygon (possibly with holes), with the shortest distance such that every point on the polygon boundary is visible from at least one point on the route. Here we consider the anchored version where the start position is given [5]. Although seemingly very related to two well-known NPhard problems, namely the Art Gallery Problem with Point Guards [13] (Point AGP) and the Euclidean Traveling Salesman Problem [15] (Euclidean TSP), WRP is solvable in polynomial time for simple polygons. It is still NP-hard for polygons with holes [4]. WRP makes impractical assumptions that the watchman senses continuously along the route (taking infinite number of viewpoints) and that the sensing actions do not incur any cost. For instance, in an environment inspection task by a robot-sensor system, each sensing action incurs a large overhead, corresponding to image acquisition, processing, and integration [17]. In addition, often for better sensing qualities, the robot has to stop its movements during image acquisitions. We introduce the problem of generalized watchman route with discrete view cost, or GWRP in short , to relax the continuous sensing assumption of WRP. It refers to planning both a route and a number of discrete viewpoints on it, such that every point on the polygon boundary is visible from at least one planned viewpoint; while the cost is minimized. The cost is a weighted sum of both view cost, proportional to the number of viewpoints planned, and the traveling cost, the total length of the route. GWRP is not a simple extension to the WRP. First, for cases where traveling cost is negligible, GWRP is reduced to Point AGP. So unlike WRP, which is in P for simple polygons, the GWRP is NP-hard. Second, as noticed in [10], the optimal WRP solution may incur an unbounded cost for the corresponding GWRP solution, i.e., infinite number of viewpoints are needed on the route to cover the whole polygon boundary. In [3, 9], the authors consider the problem of choosing a set of discrete viewpoints on a given route, while maintain-