In this project, we initially studied the Art Gallery Problem (AGP)
with vertex guards, whose goal is to minimize the number of vertex
guards required to watch an art gallery whose boundary is an n-vertex
polygon. Unlike other works that have appeared in the literature, the
objective here was to design an exact algorithm. We designed an
algorithm that iteratively computes optimal solutions to Set Cover
Problems (SCPs) corresponding to discretization of P. While it is
shown  that this procedure converges to an exact solution of
the original continuous problem, the number of iterations executed is
highly dependent on the way we discretized the polygon. Although the best
theoretical bound for convergence is O(n³) iterations, we show that, in
practice, it is achieved after a small number of them, even for random
polygons of thousands of vertices.
After 2011, we started working on the Art Gallery
Problem with Point Guards, which is a more general version of the AGP,
where the guards may be positioned at any point in the interior or on
the boundary of the input polygon. As before, we seek to find exact
solutions. An algorithm has been implemented and tested, which solves
to optimality a very large collection of polygons (with and without
holes) of multiple classes .
- M. C. Couto, P. J. de
Rezende and C. C. de Souza,
An exact algorithm
for minimizing vertex guards on art galleries, 2011, ITOR.
- D. C. Tozoni, P. J. de Rezende and C. C.
The Quest for Optimal Solutions for the Art Gallery Problem: A
Practical Iterative Algorithm, 2013, SEA.