We consider the classic problem of makespan minimization on a fixed number $n$ of machines with possibly different speeds (``uniform machines''). In an attempt to improve the makespan, we allow a fixed number $s$ of jobs to be split between two or more machines. We show that makespan minimization on $n\geq 3$ uniform machines with $s$ split jobs can be solved in polynomial time whenever $s\geq n-2$, while it is {\sf NP}-complete otherwise even for identical machines. We provide a {\it Fully Polynomial-Time Approximation Scheme} ({\sf FPTAS}) to deal with the case $s < n-2$. The main technique we use is a two-way polynomial-time reduction between makespan-minimization with splitting and a second variant, which may be of independent interest, in which the makespan must be within a pre-specified interval. We prove that, for any fixed integer $n\geq 3$, 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 job size, and it is {\sf NP}-complete otherwise even for identical machines. Using the same reduction, we implement a state-space-search algorithm for makespan minimization with any number $s$ of split jobs, and use it in computerized simulations to evaluate the effect of $s$ on the makespan.

翻译：我们考虑在固定数量$n$台可能具有不同速度（“均匀机器”）的机器上最小化完工时间的经典问题。为改善完工时间，我们允许固定数量$s$个作业在两台或多台机器之间进行分割。我们证明：当$s\geq n-2$时，在$n\geq 3$台均匀机器上通过$s个$分割作业最小化完工时间可在多项式时间内求解；而当$s<n-2$时，即便对于相同机器该问题也是{\sf NP}-完全的。针对$s<n-2$的情况，我们提出一个{\it 完全多项式时间近似方案}（{\sf FPTAS}）。主要技术手段是在带分割的完工时间最小化问题与另一个可能具有独立意义的变体之间建立双向多项式时间归约——后者要求完工时间必须落在预先指定的区间内。我们证明：对任意固定整数$n\geq 3$，若允许区间的长度至少为最大作业尺寸的$(n-2)/n$倍，则该变体可在多项式时间内求解；否则即便对于相同机器也是{\sf NP}-完全的。利用相同的归约方法，我们实现了面向任意数量$s$个分割作业的完工时间最小化状态空间搜索算法，并通过计算机仿真评估了$s$对完工时间的影响。