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

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 $1||\sum wjUj$ and $Pm||C{max}$, as well as the algorithm by Lee and Uzsoy for $P2||\sum wjCj$ are probably optimal. There is still small room for improvement for the $1|Rej\leq Q|\sum wjUj$ algorithm by Zhang et al., the algorithm for $1||\sum Tj$ by Lawler and the FPTAS for $1||\sum wjUj$ 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.