May be able to do some decomposition even with twin tiles

[bernardosulzbach]

If two disjoint components C1 and C2 are connected through twin tiles, finding an optimal solution that leaves all C1 twin tiles down, called S(C1, 0), and an optimal solution that leaves at least one C1 twin tile up, called S(C1, 1), is all the work we need to do for C1.

This seems to be true because one element from {S(C1, 0), S(C1, 1)} x {S(C2, 0), S(C2, 1)} will solve both components optimally. This happens because moves in a component can only affect the twin tiles of the other components.

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This is very much an embryo of an idea. I don't know how to prove whether or not it works.

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In time, I am unsure about this. The solution for C1 changes C2 if they are linked, so the order might matter. In the end, this might be about finding an invariant about the twin tiles.