Partitioning Problems with Splittings and Interval Targets

The $n$-way number partitioning problem is a classic problem in combinatorial optimization, with applications to diverse settings such as fair allocation and machine scheduling. All these problems are NP-hard, but various approximation algorithms are known. We consider three closely related kinds of approximations. The first two variants optimize the partition such that: in the first variant some fixed number $s$ of items can be \emph{split} between two or more bins and in the second variant we allow at most a fixed number $t$ of \emph{splittings}. The third variant is a decision problem: the largest bin sum must be within a pre-specified interval, parameterized by a fixed rational number $u$ times the largest item size. When the number of bins $n$ is unbounded, we show that every variant is strongly {\sf NP}-complete. When the number of bins $n$ is fixed, the running time depends on the fixed parameters $s,t,u$. For each variant, we give a complete picture of its running time. For $n=2$, the running time is easy to identify. Our main results consider any fixed integer $n \geq 3$. Using a two-way polynomial-time reduction between the first and the third variant, we show that $n$-way number-partitioning with $s$ split items can be solved in polynomial time if $s \geq n-2$, and it is {\sf NP}-complete otherwise. Also, $n$-way number-partitioning with $t$ splittings can be solved in polynomial time if $t \geq n-1$, and it is {\sf NP}-complete otherwise. Finally, we show that the third variant can be solved in polynomial time if $u \geq (n-2)/n$, and it is {\sf NP}-complete otherwise. Our positive results for the optimization problems consider both min-max and max-min versions. Using the same reduction, we provide a fully polynomial-time approximation scheme for the case where the number of split items is lower than $n-2$.