Online Scheduling via Gradient Descent for Weighted Flow Time Minimization

TL;DR

It is shown that the meta-algorithm achieves scalability for minimizing total weighted flow time when the residual optimum exhibits supermodularity, thereby obtaining scalable algorithms for various scheduling problems, such as matroid scheduling, generalized network flow, and generalized arbitrary speed-up curves.

Abstract

In this paper, we explore how a natural generalization of Shortest Remaining Processing Time (SRPT) can be a powerful \emph{meta-algorithm} for online scheduling. The meta-algorithm processes jobs to maximally reduce the objective of the corresponding offline scheduling problem of the remaining jobs: minimizing the total weighted completion time of them (the residual optimum). We show that it achieves scalability for minimizing total weighted flow time when the residual optimum exhibits \emph{supermodularity}. Scalability here means it is $O(1)$-competitive with an arbitrarily small speed augmentation advantage over the adversary, representing the best possible outcome achievable for various scheduling problems. Thanks to this finding, our approach does not require the residual optimum to have a closed mathematical form. Consequently, we can obtain the schedule by solving a linear program, which makes our approach readily applicable to a rich body of applications. Furthermore, by establishing a novel connection to \emph{substitute valuations in Walrasian markets}, we show how to achieve supermodularity, thereby obtaining scalable algorithms for various scheduling problems, such as matroid scheduling, generalized network flow, and generalized arbitrary speed-up curves, etc., and this is the \emph{first} non-trivial or scalable algorithm for many of them.

Figures (1)

• Figure 1: An illustration of the proof of Lemma \ref{['lem:GD-redisual']} that shows the first direction (${\mathsf{GD}}(\textbf{x}) \leq -\sum_j w_j$). The first row represents the residual schedule before GD's processing. The second row represents the schedule we consider after GD's processing for $\text{d} t$ time units. GD processes the jobs $\textbf{z}(\tau)$, $\tau \in [0, \text{d} t)$, which is colored dark grey. $\textbf{z}'$ is a schedule of the remaining sizes after processing sizes $\textbf{Z}(\text{d} t)$. Since $\textbf{z}'$ is a feasible residual solution for remaining sizes $\textbf{x}' = \textbf{x} - \textbf{Z}(\text{d} t)$, we can upper bound ${\mathsf{GD}}(\textbf{x})$ by comparing $\textbf{z}$ and $\textbf{z}'$. Because $\textbf{z}'$ is a suffix of $\textbf{z}$, shifted by $\text{d} t$ time units, each job's completion time differs by exactly $\text{d} t$ units in both schedules. This time difference translates directly to the difference in the objective function, which is the total weight of (alive) jobs.

Theorems & Definitions (91)

• Lemma 2.1: ChadhaGKM09ImMP11ImKM18
• Proposition 2.2
• Proposition 2.3
• Definition 2.4: Walrasian Market
• Definition 2.5: Demand Correspondence
• Definition 2.6: Walrasian Equilibrium
• Definition 2.7: Gross Substitutes
• Definition 2.8
• Lemma 2.9: GulS99
• Lemma 2.10