Complexity of Minimum Corridor Guarding Problems

In this paper, the complexity of minimum corridor guarding problems is discussed. These problem can be described as: given a connected orthogo-nal arrangement of vertical and horizontal line segments and a guard with unlimited visibility along a line segment, find a tree or a closed tour with minimum total length along edges of the arrangement, such that if the guard runs on the tree or on the closed tour, all line segments are visited by the guard. These problems are proved to be NP-complete. Keywords: computational complexity, computational geometry, corridor guarding, NP-complet

Approximation Algorithms for the Two-Watchman Route in a Simple Polygon

The two-watchman route problem is that of computing a pair of closed tours in an environment so that the two tours together see the whole environment and some length measure on the two tours is minimized. Two standard measures are: the minmax measure, where we want the tours where the longest of them has smallest length, and the minsum measure, where we want the tours for which the sum of their lengths is the smallest. It is known that computing a minmax two-watchman route is NP-hard for simple rectilinear polygons and thus also for simple polygons. Also, any c-approximation algorithm for the minmax two-watchman route is automatically a 2c-approximation algorithm for the minsum two-watchman route. We exhibit two constant factor approximation algorithms for computing minmax two-watchman routes in simple polygons with approximation factors 5.969 and 11.939, having running times O(n^8) and O(n^4) respectively, where n is the number of vertices of the polygon. We also use the same techniques to obtain a 6.922-approximation for the fixed two-watchman route problem running in O(n^2) time, i.e., when two starting points of the two tours are given as input.Comment: 36 pages, 14 figure