Maximizing number of selected squares in a grid

Question

We have an $n\times n$ grid of squares, each square has a non-negative integer. Two distinct squares are neighbours if they share a row or column. A selection of squares is good if every selected square has a number of selected neighbours less than or equal to its number.

Is there a known way to find the maximum number of elements in a good selection? Clearly there is a $\mathcal O(2^{n^2}n^2)$ brute force algorithm (we can test for goodness of a selection in time $\mathcal O (n^2)$ by precomputing number of selected squares in each row and column).