Maximizing the guarded boundary of an art gallery is APX-complete

Abstract

In the Art Gallery problem, given is a polygonal gallery and the goal is to guard the gallery's interior or walls with a number of guards that must be placed strategically in the interior, on walls or on corners of the gallery. Here we consider a more realistic version: exhibits now have size and may have different costs. Moreover the meaning of guarding is relaxed: we use a new concept, that of watching an expensive art item, i.e. overseeing a part of the item. The main result of the paper is that the problem of maximizing the total value of a guarded weighted boundary is APX-complete. This is shown by an appropriate gap-preserving reduction from the MAX-5-OCCURRENCE-3-SAT problem. We also show that this technique can be applied to a number of maximization variations of the art gallery problem. In particular we consider the following problems: given a polygon with or without holes and k available guards, maximize a) the length of walls guarded and b) the total cost of paintings watched or overseen. We prove that all the above problems are APX-complete. © Springer-Verlag Berlin Heidelberg 2003.