Covering an arbitrary polygon with minimum number of squares I have a problem whereby, given an arbitrary polygon with any number of points, I need to cover the whole area by a number of fixed size squares.  I can easily find a set of squares which covers the polygon but I need to ensure that I find the minimum number of squares.
Can anyone suggest a good method to do this?
Thanks in advance.
 A: I will assume you are asking for an exact cover by nonexterior squares, as in my comment.
An old paper of mine, "Covering orthogonal polygons with squares," 1988 (link here), 
addresses a special case: both the polygon edges and the square sides are parallel to coordinate axes.
Our result was subsequently improved by Bar-Yehuda and Ben-Hanoch
(Internat. Journal of Computational Geometry & Applications, Vol. 6, No. 1, 1996;
World Scientific link).
For arbitrary polygons (and arbitrary square orientations), the problem (as I have interpreted it) is not solvable, because
any acute angle cannot be covered.
For polygons with no acute angles, this problem has been extensively studied,
partly for its application to VLSI masking.
I recommend looking at
the 1997 paper,
"Approximation algorithms for covering polygons with squares and similar problems"
by Levcopoulos and Gudmundsson (Springer link).
They achieve a constant-factor approximation with a roughly quadratic algorithm.
They also provide a useful summary of related work in their Introduction.
