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Previous: Problem 55: Pallet Loading
What is the complexity of deciding whether a given number of axis-parallel unit squares can be packed into a simple polygon (without holes)?
Natural packing problem.
The problem is known to be NP-hard for polygons with holes [FPT81], even if the polygon is an orthogonal polygon with all coordinates being multiples of 1/2. Recently this version of the problem was shown to be in NP [DEKIvO09], making it NP-complete.
The problem is the decision version for two optimization problems of very different behavior. There is a PTAS for packing the maximum number of squares of fixed size [HM85]. Maximizing the size of squares such that a fixed number of squares can be packed has a lower bound on approximation of 14/13, and there is a 3/2-approximation [BF01].
What is the complexity of pallet loading? (Problem 55)
[BF01] conjecture the problem to be polynomially solvable.
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