Unifying Scheduling Algorithms for Group Completion Time
Alexander Lindermayr, Zhenwei Liu, Nicole Megow
TL;DR
The paper introduces two unifying abstractions, PSP-G and DPSP-G, for min-sum objectives over groups that generalize makespan and sum of weighted completion times. It develops a non-clairvoyant online algorithm based on Proportional Fairness achieving an O(log max_S |S|) competitive ratio, with matching lower bounds, and a versatile offline interval-LP based framework that yields (2+epsilon)-approximation guarantees for offline PSP-G and related problems. The framework is instantiated across diverse domains, delivering concrete approximation improvements for identical and related machines, preemptive data migration, and various sum coloring/sum multicoloring problems on special graph classes (perfect, interval, line graphs). The results provide a cohesive toolkit that links scheduling, graph coloring, and coflow-like problems, offering new insights and stronger guarantees in both online and offline regimes. Overall, the work advances a unified theory and practical algorithms for group-based completion-time objectives with broad applicability to systems and networks.
Abstract
We propose new abstract problems that unify a collection of scheduling and graph coloring problems with general min-sum objectives. Specifically, we consider the weighted sum of completion times over groups of entities (jobs, vertices, or edges), which generalizes two important objectives in scheduling: makespan and sum of weighted completion times. We study these problems in both online and offline settings. In the non-clairvoyant online setting, we give a novel $O(\log g)$-competitive algorithm, where $g$ is the size of the largest group. This is the first non-trivial competitive bound for many problems with group completion time objective, and it is an exponential improvement over previous results for non-clairvoyant coflow scheduling. Notably, this bound is asymptotically best-possible. For offline scheduling, we provide powerful meta-frameworks that lead to new or stronger approximation algorithms for our new abstract problems and for previously well-studied special cases. In particular, we improve the approximation ratio from $13.5$ to $10.874$ for non-preemptive related machine scheduling and from $4+\varepsilon$ to $2+\varepsilon$ for preemptive unrelated machine scheduling (MOR 2012), and we improve the approximation ratio for sum coloring problems from $10.874$ to $5.437$ for perfect graphs and from $11.273$ to $10.874$ for interval graphs (TALG 2008).