# Shared processor scheduling

## TL;DR

This paper extends existing algorithms for shared processor scheduling by providing an efficient $O(n \,\log\, n)$ algorithm for a specific class of instances and a 50% approximation for the general weighted case.

## Contribution

It introduces an $O(n \log n)$ optimization algorithm for a class of instances where job orderings relate to processing times and weights, generalizing the unweighted case.

## Key findings

- The algorithm efficiently solves a specific class of scheduling problems.
- A 50% approximation algorithm is proposed for the general weighted problem.
- The complexity of the weighted problem remains unresolved.

## Abstract

We study the shared processor scheduling problem with a single shared processor where a unit time saving (weight) obtained by processing a job on the shared processor depends on the job. A polynomial-time optimization algorithm has been given for the problem with equal weights in the literature. This paper extends that result by showing an $O(n \log n)$ optimization algorithm for a class of instances in which non-decreasing order of jobs with respect to processing times provides a non-increasing order with respect to weights --- this instance generalizes the unweighted case of the problem. This algorithm also leads to a $\frac{1}{2}$-approximation algorithm for the general weighted problem. The complexity of the weighted problem remains open.