  Given a simple polygon $\cal P$, in the Art Gallery problem the goal is to
find the minimum number of guards needed to cover the entire $\cal P$, where a
guard is a point and can see another point $q$ when $\overline{pq}$ does not
cross the edges of $\cal P$.
  This paper studies a variant of the Art Gallery problem in which guards are
restricted to lie on a dense grid inside $\cal P$. In the general problem,
guards can be anywhere inside or on the boundary of $\cal P$. The general
problem is called the \emph{point} guarding problem. It was proved that the
point guarding problem is APX-complete, meaning that we cannot do better than a
constant-factor approximation algorithm unless $P = NP$.
  A huge amount of research is committed to the studies of combinatorial and
algorithmic aspects of this problem, and as of this time, we could not find a
constant factor approximation for simple polygons. The last best-known
approximation factor for point guarding a simple polygon was $\mathcal{O}(\log
(|OPT|))$ introduced by E. Bonnet and T. Miltzow in 2020, where $|OPT|$ is the
size of the optimal solution. Here, we propose an algorithm with a constant
approximation factor for the point guarding problem where the location of
guards is restricted to a grid. The running time of the proposed algorithm
depends on the number of cells of the grid. The approximation factor is
constant regardless of the grid we use, the running time could be
super-polynomial if the grid size becomes exponential.
