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So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...

I am looking for a reference for a result that I am aware of. Let me describe the result. Given a polygon $C$ and polygons $p_1,\ldots,p_n$, it can be decided in NP time, if $p_1,\ldots,p_n$ can be ...