In our city, the waste treatment center from each ward office sends a garbage truck every other day to the houses in the area to collect garbage. On those days, we sort out garbage into recyclable, non-burnable, burnable, etc.; and some of it, like papers and plastic bottles, are recycled. Also, we help the garbage collection ward by minimizing the volume of garbage containers. For example, if we put empty boxes into a garbage bag, we should flatten them to decrease the total volume. So, I now have a common question that we should consider on a daily basis, especially like for garbage. For a given paper polyhedron P, what is the most efficient way to make it flat? That is to say, how can we minimize the total length d(P) (or simply d) of segments along which the surface of P was cut to make a net of P? A net obtained in this manner is called a net with minimum perimeter length (NMPL), or a minimum perimeter net, for short. If we represent the perimeter length of P by ℓ(P), then ℓ(P) = 2d(P) holds.
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