Computing simple paths among obstacles

AbstractGiven a set X of points in the plane, two distinguished points s,t∈X, and a set Φ of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in Φ. We present two results: (1) we show that finding such simple paths among arbitrary obstacles is NP-complete, and (2) we give a polynomial-time algorithm that computes simple paths when the obstacles form a simple polygon P and X is inside P. Our algorithm runs in time O(m2n2), where m is the number of vertices of P and n is the number of points in X

The Visibility Freeze-Tag Problem

In the Freeze-Tag Problem, we are given a set of robots at points inside some metric space. Initially, all the robots are frozen except one. That robot can awaken (or “unfreeze”) another robot by moving to its position, and once a robot is awakened, it can move and help to awaken other robots. The goal is to awaken all the robots in the shortest time. The Freeze-Tag Problem has been studied in different metric spaces: graphs and Euclidean spaces. In this thesis, we look at the Freeze-Tag Problem in polygons, and we introduce the Visibility Freeze-Tag Problem, where one robot can awaken another robot by “seeing” it. Furthermore, we introduce a variant of the Visibility Freeze-Tag Problem, called the Line/Point Freeze Tag Problem, where each robot lies on an awakening line, and one robot can awaken another robot by touching its awakening line. We survey the current results for the Freeze-Tag Problem in graphs, Euclidean spaces and polygons. Since the Visibility Freeze-Tag Problem bears some resemblance to the Watchman Route Problem, we also survey the background literature on the Watchman Route Problem. We show that the Freeze-Tag Problem in polygons and the Visibility Freeze-Tag Problem are NP-hard, and we present an O(n)-approximation algorithm for the Visibility Freeze-Tag Problem. For the Line/Point Freeze-Tag Problem, we give a polynomial time algorithm for the special case where all the awakening lines are parallel to each other. We prove that the general case is NP-hard, and we present an O(1)- approximation algorithm

Minimal link visibility paths inside a simple polygon

AbstractWe show that it is NP-hard to find a polygonal path π with a minimum number of turns inside a simple polygon P such that every point of P is visible from at least one point on π. In proving this main result, we show two other related problems to be NP-hard as well. Specifically, given a set S of points (edges) in P, the problems of finding a tour witha minimum number of turns that visits each point (edge) in S exactly once are also shown to be NP-hard. An approximation algorithm that finds a suboptimal path with the number of turns no greater than 3 times that of an optimal solution is also presented

Minimal Link Visibility Paths inside a Simple Polygon

We show that it is NP-hard to find a polygonal path π with a minimum number of turns inside a simple polygon P such that every point of P is visible from at least one point on π. In proving this main result, we show two other related problems to be NP-hard as well. Specifically, given a set S of points (edges) in P, the problems of finding a tour witha minimum number of turns that visits each point (edge) in S exactly once are also shown to be NP-hard. An approximation algorithm that finds a suboptimal path with the number of turns no greater than 3 times that of an optimal solution is also presented