Number Partitioning with Splitting

04/25/2022

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by   Samuel Bismuth, et al.

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We consider a variant of the n-way number partitioning problem, in which some fixed number s of items can be split between two or more bins. We show a two-way polynomial-time reduction between this variant and a second variant, in which the maximum bin sum must be within a pre-specified interval. We prove that the second variant can be solved in polynomial time if the length of the allowed interval is at least (n-2)/n times the maximum item size, and it is NP-hard otherwise. Using the equivalence between the variants, we prove that number-partitioning with s split items can be solved in polynomial time if s≥ n-2, and it is NP-hard otherwise.