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    An EPTAS for Scheduling on Unrelated Machines of Few Different Types

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    In the classical problem of scheduling on unrelated parallel machines, a set of jobs has to be assigned to a set of machines. The jobs have a processing time depending on the machine and the goal is to minimize the makespan, that is the maximum machine load. It is well known that this problem is NP-hard and does not allow polynomial time approximation algorithms with approximation guarantees smaller than 1.51.5 unless P==NP. We consider the case that there are only a constant number KK of machine types. Two machines have the same type if all jobs have the same processing time for them. This variant of the problem is strongly NP-hard already for K=1K=1. We present an efficient polynomial time approximation scheme (EPTAS) for the problem, that is, for any Īµ>0\varepsilon > 0 an assignment with makespan of length at most (1+Īµ)(1+\varepsilon) times the optimum can be found in polynomial time in the input length and the exponent is independent of 1/Īµ1/\varepsilon. In particular we achieve a running time of 2O(Klogā”(K)1Īµlogā”41Īµ)+poly(āˆ£Iāˆ£)2^{\mathcal{O}(K\log(K) \frac{1}{\varepsilon}\log^4 \frac{1}{\varepsilon})}+\mathrm{poly}(|I|), where āˆ£Iāˆ£|I| denotes the input length. Furthermore, we study three other problem variants and present an EPTAS for each of them: The Santa Claus problem, where the minimum machine load has to be maximized; the case of scheduling on unrelated parallel machines with a constant number of uniform types, where machines of the same type behave like uniformly related machines; and the multidimensional vector scheduling variant of the problem where both the dimension and the number of machine types are constant. For the Santa Claus problem we achieve the same running time. The results are achieved, using mixed integer linear programming and rounding techniques

    Machine Scheduling with Resource Dependent Processing Times

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    We consider several parallel machine scheduling settings with the objective to minimize the schedule makespan. The most general of these settings is unrelated parallel machine scheduling. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource. A given amount of that resource, e.g. workers, can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes classical machine scheduling problems, adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of integer programming formulations for relaxations of the respective problems, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham''s list scheduling, we thus prove the existence of constant factor approximation algorithms. Our performance guarantee is 6.83 for the most general case of unrelated parallel machine scheduling. We improve this bound for two special cases, namely to 5.83 whenever the jobs are assigned to machines beforehand, and to (5+e), e>0, whenever the processing times do not depend on the machine. Moreover, we discuss tightness of the relaxations, and derive inapproximability results.operations research and management science;
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