How to Place Efficiently Guards and Paintings in an Art Gallery

In the art gallery problem the goal is to place guards (as few as possible) in a polygon so that a maximal area of the polygon is covered. We address here a closely related problem: how to place paintings and guards in an art gallery so that the total value of guarded paintings is a maximum. More formally, a simple polygon is given along with a set of paintings. Each painting, has a length and a value. We study how to place at the same time: i) a given number of guards on the boundary of the polygon and ii) paintings on the boundary of the polygon so that the total value of guarded paintings is maximum. We investigate this problem for a number of cases depending on: i) where the guards can be placed (vertices, edges), ii) whether the polygon has holes or not and iii) whether the goal is to oversee the placed paintings (every point of a painting is seen by at least one guard), or to watch the placed paintings (at least one point of a painting is seen by at least one guard). We prove that the problem is NP-hard in all the above cases and we present polynomial time approximation algorithms for all cases, achieving constant ratios.