Online Scheduling via Gradient Descent for Weighted Flow Time Minimization

TL;DR

This paper introduces a gradient descent-based meta-algorithm for online scheduling that achieves near-optimal weighted flow time minimization by leveraging supermodularity and linear programming, applicable to diverse complex scheduling problems.

Contribution

It proposes a novel meta-algorithm for online scheduling that is scalable and does not require closed-form residual optima, connecting to market theory for broader applicability.

Findings

Achieves $O(1)$-competitiveness with small speed augmentation.

Extends to various scheduling problems via supermodularity.

First scalable algorithms for complex scheduling scenarios.

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.