The paper considers the assignment of items to groups according to their attribute values such that the groups are as balanced as possible. Although the problem is in general NP-hard, we prove that it can be solved in pseudo-polynomial time if attribute values are integer. We point out a relation to partition and more general to multi-way number partitioning. Furthermore, we introduce a mixed-integer programming (MIP) formulation, a variable reduction technique, and an efficient lower bound for the objective value. Our computational results show that the lower bound meets the optimal objective value in the most of our instances of realistic size. Hence, the MIP solves instances with several thousand items within seconds to optimality.

本文考虑根据项目的属性值将项目分配给组，以使组尽可能平衡。尽管该问题一般是NP难问题，但我们证明如果属性值为整数，则可以在伪多项式时间内解决该问题。我们指出与分区的关系，以及更一般的多路数字分区的关系。此外，我们引入了混合整数规划（MIP）公式、变量缩减技术和目标值的有效下界。我们的计算结果表明，在大多数实际大小的实例中，下限满足最佳目标值。因此，MIP 可以在几秒钟内解决具有数千个项目的实例，从而达到最优。