On the Complexity of Scheduling Problems With a Fixed Number of Parallel Identical Machines

TL;DR

This paper establishes conditional lower bounds for various fixed-machine scheduling problems, demonstrating the near-optimality of classical algorithms and guiding future research directions.

Contribution

It provides lower bounds and improved algorithms for several scheduling problems, clarifying the optimality of existing methods and identifying areas for further improvement.

Findings

Lawler and Moore's algorithm for 1||∑w_jU_j and Pm||C_max is probably optimal.

Lee and Uzsoy's algorithm for P2||∑w_jC_j is probably optimal.

Improved dynamic programming algorithms for P2|any|C_max and P3|any|C_max.

Abstract

In parallel machine scheduling, we are given a set of jobs, together with a number of machines and our goal is to decide for each job, when and on which machine(s) it should be scheduled in order to minimize some objective function. Different machine models, job characteristics and objective functions result in a multitude of scheduling problems and many of them are NP-hard, even for a fixed number of identical machines. In this work, we give conditional running time lower bounds for a large number of scheduling problems, indicating the optimality of some classical algorithms. Most notably, we show that the algorithm by Lawler and Moore for $1||\sum w_jU_j$ and $Pm||C_{max}$, as well as the algorithm by Lee and Uzsoy for $P2||\sum w_jC_j$ are probably optimal. There is still small room for improvement for the $1|Rej\leq Q|\sum w_jU_j$ algorithm by Zhang et al., the algorithm for $1||\sum T_j$ by Lawler and the FPTAS for $1||\sum w_jU_j$ by Gens and Levner. We also give a lower bound for $P2|any|C_{max}$ and improve the dynamic program by Du and Leung from $\mathcal{O}(nP^2)$ to $\mathcal{O}(nP)$, matching this new lower bound. Here, $P$ is the sum of all processing times. The same idea also improves the algorithm for $P3|any|C_{max}$ by Du and Leung from $\mathcal{O}(nP^5)$ to $\mathcal{O}(nP^2)$. While our results suggest the optimality of some classical algorithms, they also motivate future research in cases where the best known algorithms do not quite match the lower bounds.