The Continuous 1.5{D} Terrain Guarding Problem: {D}iscretization, Optimal Solutions, and {PTAS}

In the NP-hard continuous 1.5D Terrain Guarding Problem (TGP) we are given an x-monotone chain of line segments in the plain (the terrain $T$), and ask for the minimum number of guards (located anywhere on $T$) required to guard all of $T$. We construct guard candidate and witness sets $G, W \subset T$ of polynomial size, such that any feasible (optimal) guard cover $G' \subseteq G$ for $W$ is also feasible (optimal) for the continuous TGP. This discretization allows us to: (1) settle NP-completeness for the continuous TGP; (2) provide a Polynomial Time Approximation Scheme (PTAS) for the continuous TGP using the existing PTAS for the discrete TGP by Gibson et al.; (3) formulate the continuous TGP as an Integer Linear Program (IP). Furthermore, we propose several filtering techniques reducing the size of our discretization, allowing us to devise an efficient IP-based algorithm that reliably provides optimal guard placements for terrains with up to 1000000 vertices within minutes on a standard desktop computer

A finite dominating set of cardinality O(k) and a witness set of cardinality O(n) for 1.5D terrain guarding problem

1.5 dimensional (1.5D) terrain is characterized by a piecewise linear curve. Locating minimum number of guards on the terrain (T) to cover/guard the whole terrain is known as 1.5D terrain guarding problem. Approximation algorithms and a polynomial-time approximation scheme have been presented for the problem. The problem has been shown to be NP-Hard. In the problem, the set of possible guard locations and the set of points to be guarded are uncountable. To solve the problem to optimality, a finite dominating set (FDS) of size O (n2) and a witness set of size O (n3) have been presented, where n is the number of vertices on T. We show that there exists an even smaller FDS of cardinality O (k) and a witness set of cardinality O(n), where k is the number of convex points. Convex points are vertices with the additional property that between any two convex points the piecewise linear curve representing the terrain is convex. Since it is always true that k≤ n for n≥ 2 and since it is possible to construct terrains such that n= 2 k, the existence of an FDS with cardinality O(k) and a witness set of cardinality of O (n) leads to the reduction of decision variables and constraints respectively in the zero-one integer programming formulation of the problem. © 2017, Springer Science+Business Media New York

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

Parameter Analysis for Guarding Terrains

The Terrain Guarding problem is a well-known variant of the famous Art Gallery problem. Only second to Art Gallery, it is the most well-studied visibility problem in Discrete and Computational Geometry, which has also attracted attention from the viewpoint of Parameterized complexity. In this paper, we focus on the parameterized complexity of Terrain Guarding (both discrete and continuous) with respect to two natural parameters. First we show that, when parameterized by the number r of reflex vertices in the input terrain, the problem has a polynomial kernel. We also show that, when parameterized by the number c of minima in the terrain, Discrete Orthogonal Terrain Guarding has an XP algorithm