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Consider a set of jobs connected to a directed acyclic task graph with a fixed source and sink. The edges of this graph model precedence constraints and the jobs have to be scheduled with respect to those. We introduce the server cloud scheduling problem, in which the jobs have to be processed either on a single local machine or on one of infinitely many cloud machines. For each job, processing times both on the server and in the cloud are given. Furthermore, for each edge in the task graph, a communication delay is included in the input and has to be taken into account if one of the two jobs is scheduled on the server and the other in the cloud. The server processes jobs sequentially, whereas the cloud can serve as many as needed in parallel, but induces costs. We consider both makespan and cost minimization. The main results are an FPTAS for the makespan objective for graphs with a constant source and sink dividing cut and strong hardness for the case with unit processing times and delays. [ABSTRACT FROM AUTHOR]

Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items has to be placed in multiple target locations. Herein, a configuration describes a possible placement on one of the target locations, and the IP is used to choose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and, therefore, be solved efficiently. As an application, we consider scheduling problems with setup times in which a set of jobs has to be scheduled on a set of identical machines with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed, an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time f (1 / ε) · poly (| I |) . Previously, only constant factor approximations of 5/3 and 4 / 3 + ε , respectively, were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine. [ABSTRACT FROM AUTHOR]

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 1.5 unless P = NP. We consider the case that there are only a constant number K 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 = 1 . We present an efficient polynomial time approximation scheme (EPTAS) for the problem, that is, for any ε > 0 an assignment with makespan of length at most (1 + ε) times the optimum can be found in polynomial time in the input length and the exponent is independent of 1 / ε . In particular, we achieve a running time of 2 O (K log (K) 1 ε log 4 1 ε) + poly (| I |) , where |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. [ABSTRACT FROM AUTHOR]

We consider a special case of the scheduling problem on unrelated machines, namely the restricted assignment problem with two different processing times. We show that the configuration LP has an integrality gap of at most 53≈1.667 for this problem. This allows us to estimate the optimal makespan within a factor of 53, improving upon the previously best known estimation algorithm with ratio 116≈1.833 due to Chakrabarty et al. (in: Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms (SODA 2015), pp 1087-1101, 2015). [ABSTRACT FROM AUTHOR]