On the Complexity of Scheduling Problems With a Fixed Number of Parallel Identical Machines
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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. However, using pseudo-polynomial or approximation algorithms, we can still hope to solve some of these problems efficiently. In this work, we give conditional running time lower bounds for a large number of scheduling problems, indicating the optimality of some classical algorithms. In particular, we show that the dynamic programming algorithm by Lawler and Moore is probably optimal for 1||∑ w_jU_j and Pm||C_max. Moreover, the algorithm by Lee and Uzsoy for P2||∑ w_jC_j is probably optimal as well. There is still small room for improvement for the 1|Rej≤ Q|∑ w_jU_j algorithm by Zhang et al., the algorithm for 1||∑ T_j by Lawler and the FPTAS for 1||∑ 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 𝒪(nP^2) to 𝒪(nP) to match this 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 𝒪(nP^5) to 𝒪(nP^2). The lower bounds in this work all either rely on the (Strong) Exponential Time Hypothesis or the (min,+)-conjecture. 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.
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