# Covering an arbitrary polygon with minimum number of squares

I have a problem whereby, given an arbitrary polyogn 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?

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I think you need to sharpen the description of your problem: one large square can cover any polygon. Perhaps you mean: the union of the squares must be exactly the polygon, but no square can contain a point exterior to the polygon? – Joseph O'Rourke Mar 15 '12 at 13:50
I agree with Joseph. Is the size of the squares fixed before the polygon is given? Are you allowing the squares to be rotated, or must they have sides parallel to the axes? – Lee Mosher Mar 15 '12 at 13:53
Apologies for the lack of clarity on my question. The polygon I wish to cover can be any size and shape. My squares are always smaller than the main polygon and of a fixed size. Therefore one bounding square will not work. The squares must also be upright (i.e. they cant be rotated). The union of all my squares must cover the completely to main polygon and I'm looking to find the minimum number of squares. The union of squares can include points exterior to the polygon. – Chris Mar 15 '12 at 15:37
There remains one additional under-specified aspect: May the squares overlap one another? Or are they to be arranged in a grid? – Joseph O'Rourke Mar 16 '12 at 0:40
They can overlap. – Chris Mar 17 '12 at 21:59