Online Makespan Scheduling with Job Migration on Uniform Machines

In the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and $$\approx 1.4659$$ ≈ 1.4659 . They show that $$k = O(m)$$ k = O ( m ) is sufficient to achieve this bound and no $$k = o(n)$$ k = o ( n ) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a $$\delta = \varTheta (1)$$ δ = Θ ( 1 ) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than $$1.4659 + \delta $$ 1.4659 + δ with $$k = o(n)$$ k = o ( n ) . We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and $$\approx 1.7992$$ ≈ 1.7992 with $$k = O(m)$$ k = O ( m ) . We also show that $$k = \varOmega (m)$$ k = Ω ( m ) is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines.

Branch Less, Cut More and Schedule Jobs with Release and Delivery Times on Uniform Machines

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.

A Polynomial Algorithm for Sequencing Jobs with Release and Delivery Times on Uniform Machines

The problem of sequencing $n$ equal-length non-simultaneously released jobs with delivery times on $m$ uniform machines to minimize the maximum job completion time is considered. To the best of our knowledge, the complexity status of this classical scheduling problem remained open up to the date. We establish its complexity status positively by showing that it can be solved in polynomial time. We adopt for the uniform machine environment the general algorithmic framework of the analysis of behavior alternatives developed earlier for the identical machine environment. The proposed algorithm has the time complexity $O(\gamma m^2 n\log n)$, where $\gamma$ can be any of the magnitudes $n$ or $q_{\max}$, the maximum job delivery time. In fact, $n$ can be replaced by a smaller magnitude $\kappa&lt;n$, which is the number of special types of jobs (it becomes known only upon the termination of the algorithm).

A Competitive Online Algorithm for Minimizing Total Weighted Completion Time on Uniform Machines

We consider the classic online scheduling problem on m uniform machines in the online setting where jobs arrive over time. Preemption is not allowed. The objective is to minimize total weighted completion time. An online algorithm based on the directly waiting strategy is proposed. Its competitive performance is proved to be max2smax1−1/2∑si,2smax/1+smax2.5−1/2m by the idea of instance reduction, where sm is the fastest machine speed after being normalized by the slowest machine speed.