Abstract
We consider the problem of scheduling n jobs with identical processing times and given release as well as delivery times on m uniform machines. The goal is to minimize the makespan, i.e., the maximum full completion time of any job. This problem is well-known to have an open complexity status even if the number of jobs is fixed. We present a polynomial-time algorithm for the problem which is based on the earlier introduced algorithmic framework blesscmore (“branch less and cut more”). We extend the analysis of the so-called behavior alternatives developed earlier for the version of the problem with identical parallel machines and show how the earlier used technique for identical machines can be extended to the uniform machine environment if a special condition on the job parameters is imposed. The time complexity of the proposed algorithm is O(γm2nlogn), where γ can be either n or the maximum job delivery time qmax. This complexity can even be reduced further by using a smaller number κ<n in the estimation describing the number of jobs of particular types. However, this number κ becomes only known when the algorithm has terminated.
Highlights
In this paper, we consider a basic optimization problem of scheduling jobs with release and delivery times on uniform machines with the objective to minimize the makespan.More precisely, n jobs from the set J = {1, 2, ..., n} are to be processed by m parallel uniform machines from the set M = {1, 2, ..., m}
We showed that the earlier developed technique for scheduling identical machines can be extended to the uniform machine environment if Condition (1) on the job parameters is satisfied, making a step towards the settlement of the complexity status of this longstanding open problem
The imposed condition reflects potential conflicts that arise in the uniform machine environment but do not arise in the identical machine environment
Summary
We consider a basic optimization problem of scheduling jobs with release and delivery times on uniform machines with the objective to minimize the makespan. The studied multiprocessor optimization problem, described below, is commonly abbreviated as Q| p j =p, r j , q j |Cmax (its version with identical parallel machines is abbreviated as P| p j =p, r j , q j |Cmax , the first field specifies the machine environment, the second one the job parameters, and the third one the objective function) It is well-known that there is an equivalent (perhaps more traditional) formulation of the above described problem, in which, instead of the delivery time q j , every job j has its due-date d j. We present a polynomial-time algorithm for the uniform machine environment which finds an optimal solution to the problem if for any pair of jobs i and j with qi > q j and r j > ri , we have qi − q j ≥ r j − ri (1).
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