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Maximizing the Guarded Interior of an Art Gallery
 in: Proc. 22nd European Workshop on Computational Geometry
"... In the Art Gallery problem a polygon is given and the goal is to place as few guards as possible so that the entire area of the polygon is covered. We address a closely related problem: how to place a fixed number of guards on the vertices or the edges of a simple polygon so that the total guarded a ..."
Abstract

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In the Art Gallery problem a polygon is given and the goal is to place as few guards as possible so that the entire area of the polygon is covered. We address a closely related problem: how to place a fixed number of guards on the vertices or the edges of a simple polygon so that the total guarded area inside the polygon is maximized. Recall that an optimization problem is called APXhard, if there exists an ɛ> 0 such that an approximation ratio of 1 + ɛ cannot be guaranteed by any polynomial time approximation algorithm, unless P = NP. We prove that our problem is APXhard and we present a polynomial time algorithm achieving constant approximation ratio. Finally we extend our results for the case where the guards are required to cover valued items inside the polygon. The valued items or “treasures ” are modeled as simple closed polygons. 1
Lower bounds on the obstacle number of graphs ∗
"... Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if and only if the corresponding points can be connected by a segment which avoids all obstacl ..."
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Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if and only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of G is the minimum number of obstacles in an obstacle representation of G. It is shown that there are graphs on n vertices with obstacle number at least Ω(n/logn).