We propose new abstract and unified perspectives on a range of scheduling and graph coloring problems with general min-sum objectives. Specifically, we consider various problems where the objective function is the weighted sum of completion times over groups of entities (jobs, vertices, or edges), thereby generalizing two important objectives in scheduling: makespan and the sum of weighted completion times. As one of our main results, we present a best-possible $\mathcal O(\log g)$-competitive algorithm in the non-clairvoyant online setting, where $g$ denotes the size of the largest group. This is the first non-trivial competitive bound for several problems with group completion time objective, and it is an exponential improvement over previous results for non-clairvoyant coflow scheduling. For offline scheduling, we provide elegant yet powerful meta-frameworks that, in a unifying way, yield new or stronger approximation algorithms for our new abstract problems as well as for previously well-studied special cases.

翻译：我们针对一类具有广义最小和目标的调度与图着色问题，提出了新颖且统一的抽象视角。具体而言，我们研究了目标函数为实体组（作业、顶点或边）完成时间加权和的多种问题，从而推广了调度中两个重要目标：完工时间和加权完成时间之和。作为主要成果之一，我们在非先知在线设定下提出了一个最优的 $\mathcal O(\log g)$-竞争算法，其中 $g$ 表示最大组的规模。这是针对具有组完成时间目标的若干问题首次获得非平凡的竞争界，且相较于先前非先知协同流调度的结果实现了指数级改进。对于离线调度，我们提出了简洁而强大的元框架，以统一的方式为新的抽象问题以及先前已深入研究的特例，推导出新的或更强的近似算法。