Partitioning Problems with Splitting and Interval Targets

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 $s\geq n-2$, and it is NP-hard otherwise.