  The art gallery problem enquires about the least number of guards that are
sufficient to ensure that an art gallery, represented by a polygon $P$, is
fully guarded. In 1998, the problems of finding the minimum number of point
guards, vertex guards, and edge guards required to guard $P$ were shown to be
APX-hard by Eidenbenz, Widmayer and Stamm. In 1987, Ghosh presented
approximation algorithms for vertex guards and edge guards that achieved a
ratio of $\mathcal{O}(\log n)$, which was improved upto $\mathcal{O}(\log\log
OPT)$ by King and Kirkpatrick in 2011. It has been conjectured that
constant-factor approximation algorithms exist for these problems. We settle
the conjecture for the special class of polygons that are weakly visible from
an edge and contain no holes by presenting a 6-approximation algorithm for
finding the minimum number of vertex guards that runs in $\mathcal{O}(n^2)$
time. On the other hand, for weak visibility polygons with holes, we present a
reduction from the Set Cover problem to show that there cannot exist a
polynomial time algorithm for the vertex guard problem with an approximation
ratio better than $((1 - \epsilon)/12)\ln n$ for any $\epsilon>0$, unless NP=P.
We also show that, for the special class of polygons without holes that are
orthogonal as well as weakly visible from an edge, the approximation ratio can
be improved to 3. Finally, we consider the Point Guard problem and show that it
is NP-hard in the case of polygons weakly visible from an edge.
