Abstract
A polyhedronP = {x∈ ℝ n+ ∣d ⩽Ax ⩽e} is said to have the real decomposition property (RDP) if for any positiveT and any realx∈TP, there are positive coefficients λ1,…, λ r and integers 1, …,s r ∈P withx = λ l s l + … + λ r s r andT=λ l + ⋯ + λ r . We give a constructive proof that this property holds for a polyhedronP iffP is integral. This construction is used to show that some classes of polyhedra have an integral decomposition property. Furthemore, the RDP provides a generalization of the theorem of Birkhoff-von Neumann. So RDP may be used in some scheduling problems on parallel processors with preemptions.
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de Werra, D. A decomposition property of polyhedra. Mathematical Programming 30, 261–266 (1984). https://doi.org/10.1007/BF02591932
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DOI: https://doi.org/10.1007/BF02591932