The Complexity of Guarding Monotone Polygons

Abstract A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersection with P . A set G of points inside P or on the boundary of P is said to guard the polygon if every point inside P or on the boundary of P is seen by a point in G. An interior guard can lie anywhere inside or on the boundary of the polygon. Using a reduction from Monotone 3SAT, we prove that interior guarding a monotone polygon is NP-hard. Because interior guards can be placed anywhere inside the polygon, a clever gadget is introduced that forces interior guards to be placed at very specific locations

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

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

Optimal Guarding of Polygons and Monotone Chains (Extended Abstract)

) Danny Z. Chen Vladimir Estivill-Castro y Jorge Urrutia z Abstract In this paper we study several problems concerning the guarding of a polygon or a x- monotone polygonal chain P with n vertices from a set of points lying on it. Our results are: (1) An O(n log n) time sequential algorithm for computing the shortest guarding boundary chain of a polygon P . (2) An O(n log n) time sequential algorithm for computing the smallest set of consecutive edges guarding a polygon P . (3) Parallel algorithms for each of the two previous problems that run in O(logn) time using O(n) processors in the CREW-PRAM computational model. (4) A linear sequential algorithm for computing the smallest left-guarding set of vertices of an x-monotone polygonal chain P . (5) An optimal \Theta(n log n) sequential algorithm for computing the smallest guarding set of relays of an x-monotone polygonal chain P . (6) Finally, we consider the problem of finding the problem of placing on a x-monotone polygonal cha..