Guarding Art Galleries: The Extra Cost for Sculptures is Linear

Description

Art gallery problems have been extensively studied over the last decade and have found different type of applications. Normally the number of sides of a polygon or the general shape of the polygon is used as a measure of the complexity of the problem. In this paper we explore another measure of complexity, namely, the number of guards required to guard the boundary, or the walls, of the gallery. We prove that if n guards are necessary to guard the walls of an art gallery, then an additional team of at most 4n − 6 will guard the whole gallery. This result improves a previously known quadratic bound, and is a step towards a possibly optimal value of n − 2 additional guards. The proof is algorithmic, uses ideas from graph theory, and is mainly based on the definition of a new reduction operator which recursively eliminates the simple parts of the polygon. We also prove that every gallery with c convex vertices can be guarded by at most 2c − 4 guards, which is optimal.

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