Fixed Order Scheduling with Deadlines

2024-12-14

Operations Research Letters

AI Summary

This paper studies the deadline-constrained fixed-order scheduling problem on identical parallel machines: given $n$ jobs, each with a processing time and a deadline, the objective is to minimize the number of machines required while respecting both deadline constraints and a prescribed precedence order. We first prove that the First-Fit algorithm is optimal for the unit-processing-time case—a novel result. For four structured input classes—including monotone deadlines and ordered processing times—we design 2-approximation algorithms. For the general case, we propose the first $O(log n)$-approximation algorithm, substantially improving upon prior approaches. Our techniques integrate combinatorial optimization modeling, rigorous analysis of greedy strategies, and a novel characterization of the interplay between job slackness and deadline ordering—yielding a significant breakthrough in approximation ratio for this problem.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Minimize machines while meeting deadlines and fixed job order
• First-fit algorithm optimal for unit processing times
• O(log n)-approximation for general scheduling problem

Innovation

• First-fit algorithm for optimal scheduling
• Handles four specific order alignment cases
• Provides O(log n)-approximation for general problem