Fixed Order Scheduling with Deadlines

TL;DR

This paper investigates scheduling on identical parallel machines with a fixed global processing order on each machine, under common release times and deadlines, aiming to minimize the number of machines. It analyzes two greedy strategies, establishing that Next-Fit can perform arbitrarily badly, while First-Fit provides optimality for unit processing times and a 2-approximation in several natural order-structures, with a general $O(\log n)$-approximation for arbitrary orders. The results delineate when simple greedy schedules suffice and provide a framework—via slacks, deadlines, and a set-cover reduction—for approximating fixed-order SRDM variants. The findings have practical relevance for energy-aware scheduling and fixed-priority systems and point to future work on tighter bounds and additional algorithms.

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.

Figures (6)

• Figure 1: The optimal schedule and the schedule by NF for $\mathcal{I}_n$ where $n\geq 5$.
• Figure 2: Exchanging the jobs between two machines of $\tau^*$ when $s = s^*$.
• Figure 3: Exchanging the jobs between two machines of $\tau^*$ when $s > s^*$.
• Figure 4: An example with the approximation ratio of $(2k+1)/(k+1).$
• Figure 5: An example of the placement of jobs $e^*_i$ in the OPT and FF schedules.

Theorems & Definitions (20)

• lemma 1
• theorem 1
• Claim 1
• Claim 2
• lemma 2
• theorem 2
• Claim 3
• Claim 4
• theorem 3
• theorem 4