  The art gallery problem enquires about the least number of guards sufficient
to ensure that an art gallery, represented by a simple polygon $P$, is fully
guarded. Most standard versions of this problem are known to be NP-hard. In
1987, Ghosh provided a deterministic $\mathcal{O}(\log n)$-approximation
algorithm for the case of vertex guards and edge guards in simple polygons. In
the same paper, Ghosh also conjectured the existence of constant ratio
approximation algorithms for these problems. We present here three
polynomial-time algorithms with a constant approximation ratio for guarding an
$n$-sided simple polygon $P$ using vertex guards. (i) The first algorithm, that
has an approximation ratio of 18, guards all vertices of $P$ in
$\mathcal{O}(n^4)$ time. (ii) The second algorithm, that has the same
approximation ratio of 18, guards the entire boundary of $P$ in
$\mathcal{O}(n^5)$ time. (iii) The third algorithm, that has an approximation
ratio of 27, guards all interior and boundary points of $P$ in
$\mathcal{O}(n^5)$ time. Further, these algorithms can be modified to obtain
similar approximation ratios while using edge guards. The significance of our
results lies in the fact that these results settle the conjecture by Ghosh
regarding the existence of constant-factor approximation algorithms for this
problem, which has been open since 1987 despite several attempts by
researchers. Our approximation algorithms exploit several deep visibility
structures of simple polygons which are interesting in their own right.
