  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, and we provide a fully polynomial-time
approximation scheme for the case where the number of split items is lower than
$n-2$.
