Exact Algorithms for Terrain Guarding

Given a 1.5-dimensional terrain T, also known as an x-monotone polygonal chain, the Terrain Guarding problem seeks a set of points of minimum size on T that guards all of the points on T. Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T. The Terrain Guarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm [SODA 2005]. However, only in 2010 King and Krohn [SODA 2010] finally showed that Terrain Guarding is NP-hard. In spite of the remarkable developments in approximation algorithms for Terrain Guarding, next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this paper, we answer the first question affirmatively by developing an n^O(sqrt{k})-time algorithm for both Discrete Terrain Guarding and Continuous Terrain Guarding. We also make non-trivial progress with respect to the second question: we show that Discrete Orthogonal Terrain Guarding, a well-studied special case of Terrain Guarding, is fixed-parameter tractable

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

A (7/2)-Approximation Algorithm for Guarding Orthogonal Art Galleries with Sliding Cameras

Consider a sliding camera that travels back and forth along an orthogonal line segment $s$ inside an orthogonal polygon $P$ with $n$ vertices. The camera can see a point $p$ inside $P$ if and only if there exists a line segment containing $p$ that crosses $s$ at a right angle and is completely contained in $P$. In the minimum sliding cameras (MSC) problem, the objective is to guard $P$ with the minimum number of sliding cameras. In this paper, we give an $O(n^{5/2})$-time $(7/2)$-approximation algorithm to the MSC problem on any simple orthogonal polygon with $n$ vertices, answering a question posed by Katz and Morgenstern (2011). To the best of our knowledge, this is the first constant-factor approximation algorithm for this problem.Comment: 11 page

Patrolling a Street Network is Strongly NP-Complete but in P for Tree Structures

We consider the following problem: Given a finite set of straight line segments in the plane, determine the positions of a minimal number of points on the segments, from which guards can see all segments. This problem can be interpreted as looking for a minimal number of locations of policemen, guards, cameras or other sensors, that can observe a network of streets, corridors, tunnels, tubes, etc. We show that the problem is strongly NP-complete even for a set of segments with a cubic graph structure, but in P for tree structures