Minimum exact cover of grid with squares; extra cuts

Question

This problem appeared in a challenge, but since it is now closed it should be OK to ask about it.

The problem (not this question itself, this is just background info

Answer 1

I used the branch and price method to good effect on a similar problem (ITA's Strawberry Fields; scroll down). Here is my code, which (lacking comments) is probably only good as a zero-knowledge proof that I know what I'm talking about. It was orders of magnitude faster than a commercial solver (which I won't name).

The key was the branching strategy. Rather than branch directly on the x_i variables, which is likely what your solver is doing, branch on a higher-level decision. The one that I used for Strawberry Fields was to decide whether two cells would be covered by the same square. If you target the pairs that the fractional solution is most on the fence about, then the structure of the solution seems to be set fairly quickly.

Unfortunately, I can't offer you advice on how to program this into an existing integer program solver. For Strawberry Fields, I went with custom everything, mostly because I wanted to but in part because I was generating columns on the fly, using cumulative grid row and grid column sums to evaluate rectangles quickly.