Fixed Order Scheduling with Deadlines

TL;DR

This paper investigates a fixed order scheduling problem on parallel machines with deadlines, demonstrating optimality and approximation bounds for the first-fit algorithm under various conditions, and providing an $O( ext{log } n)$-approximation for the general case.

Contribution

It establishes the optimality of first-fit for unit processing times and provides approximation guarantees in specific scenarios, along with an $O( ext{log } n)$-approximation for the general problem.

Findings

First-fit is optimal with unit processing times.

First-fit is a 2-approximation under certain order conditions.

An $O( ext{log } n)$-approximation algorithm is proposed for the general problem.

Abstract

This paper studies a scheduling problem in a parallel machine setting, where each machine must adhere to a predetermined fixed order for processing the jobs. Given $n$ jobs, each with processing times and deadlines, we aim to minimize the number of machines while ensuring deadlines are met and the fixed order is maintained. We show that the first-fit algorithm solves the problem optimally with unit processing times and is a 2-approximation in the following four cases: (1) the order aligns with non-increasing slacks, (2) the order aligns with non-decreasing slacks, (3) the order aligns with non-increasing deadlines, and (4) the optimal solution uses at most 3 machines. For the general problem we provide an $O(\log n)$-approximation.