# Select and Permute: An Improved Online Framework for Scheduling to Minimize Weighted Completion Time

## TL;DR

This paper presents a new online scheduling framework that leverages offline approximation algorithms to achieve improved competitive ratios for minimizing weighted completion time across various complex models.

## Contribution

It introduces a novel online framework based on offline algorithms, enhancing competitive ratios for multiple scheduling problems and including a randomized variant for further improvements.

## Key findings

- Achieved best known competitive ratios for concurrent open shop, coflow, and cluster models.
- Developed a randomized framework that further improves competitive ratios.
- Demonstrated the framework's versatility across different scheduling scenarios.

## Abstract

In this paper, we introduce a new online scheduling framework for minimizing total weighted completion time in a general setting. The framework is inspired by the work of Hall et al. [Mathematics of Operations Research, Vol 22(3):513-544, 1997] and Garg et al. [Proc. of Foundations of Software Technology and Theoretical Computer Science, pp. 96-107, 2007], who show how to convert an offline approximation to an online scheme. Our framework uses two offline approximation algorithms (one for the simpler problem of scheduling without release times, and another for the minimum unscheduled weight problem) to create an online algorithm with provably good competitive ratios. We illustrate multiple applications of this method that yield improved competitive ratios. Our framework gives algorithms with the best or only known competitive ratios for the concurrent open shop, coflow, and concurrent cluster models. We also introduce a randomized variant of our framework based on the ideas of Chakrabarti et al. [Proc. of International Colloquium on Automata, Languages, and Programming, pp. 646-657, 1996] and use it to achieve improved competitive ratios for these same problems.