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