In this paper we study the relation of two fundamental problems in scheduling and fair allocation: makespan minimization on unrelated parallel machines and max-min fair allocation, also known as the Santa Claus problem. For both of these problems the best approximation factor is a notorious open question; more precisely, whether there is a better-than-2 approximation for the former problem and whether there is a constant approximation for the latter. While the two problems are intuitively related and history has shown that techniques can often be transferred between them, no formal reductions are known. We first show that an affirmative answer to the open question for makespan minimization implies the same for the Santa Claus problem by reducing the latter problem to the former. We also prove that for problem instances with only two input values both questions are equivalent. We then move to a special case called ``restricted assignment'', which is well studied in both problems. Although our reductions do not maintain the characteristics of this special case, we give a reduction in a slight generalization, where the jobs or resources are assigned to multiple machines or players subject to a matroid constraint and in addition we have only two values. This draws a similar picture as before: equivalence for two values and the general case of Santa Claus can only be easier than makespan minimization. To complete the picture, we give an algorithm for our new matroid variant of the Santa Claus problem using a non-trivial extension of the local search method from restricted assignment. Thereby we unify, generalize, and improve several previous results. We believe that this matroid generalization may be of independent interest and provide several sample applications.

翻译：本文研究了调度与公平分配中两个基本问题之间的关系：无关并行机上的最小化最大完工时间问题（makespan minimization）与最大化最小公平分配问题（max-min fair allocation，又称“圣诞老人问题”）。对于这两个问题，最佳近似比均为长期悬而未决的公开难题：具体而言，前者是否存在优于2的近似算法，后者是否存在常数近似算法。尽管这两个问题直观上相互关联，且历史表明技术手段常可在两者间迁移，但目前尚无形式化归约。我们首先证明，若最小化最大完工时间问题的公开难题得到肯定回答，则通过将圣诞老人问题归约至前者，该结论同样适用于圣诞老人问题。此外，我们证明对于仅含两个输入值的实例，这两个问题等价。随后，我们转向称为“限制分配”（restricted assignment）的特殊情形，该情形在两个问题中均被广泛研究。尽管我们的归约未保持该特殊情形的特性，但在轻微推广下（即作业或资源在拟阵约束下分配给多台机器或多个玩家，且仅含两个值的情况）给出了归约。这呈现了类似的图景：两值情形下等价，而圣诞老人问题的一般情形仅可能比最小化最大完工时间问题更易。为完善图景，我们通过非平凡地扩展限制分配情形的局部搜索方法，为圣诞老人问题的拟阵变体设计了算法。由此统一、推广并改进了若干先前结果。我们认为该拟阵推广可能具有独立研究价值，并提供了若干示例应用。