A best online algorithm for scheduling on two parallel batch machines

AbstractWe consider the online scheduling on two parallel batch machines with infinite batch size to minimize makespan, where jobs arrive over time. That is, all information of a job is not available until it is released. For this online scheduling problem, Nong et al. [Q.Q. Nong, T.C.E. Cheng, C.T. Ng, An improved online algorithm for scheduling on two unrestrictive parallel batch processing machines, Operations Research Letters, 36 (2008) 584–588] have provided an online algorithm with competitive ratio no greater than 2. We show that this bound is tight for the problem. Furthermore we give a new best possible online algorithm with a tighter structure

An asymptotically optimal online algorithm to minimize the total completion time on two multipurpose machines with unit processing times

AbstractIn the majority of works on online scheduling on multipurpose machines the objective is to minimize the makespan. We, in contrast, consider the objective of minimizing the total completion time. For this purpose, we analyze an online-list scheduling problem of n jobs with unit processing times on a set of two machines working in parallel. Each job belongs to one of two sets of job types. Jobs belonging to the first set can be processed on either of the two machines while jobs belonging to the second set can only be processed on the second machine. We present an online algorithm with a competitive ratio of ρLB+O(1n), where ρLB is a lower bound on the competitive ratio of any online algorithm and is equal to 1+(−α+4α3−α2+2α−12α2+1)2 where α=13+16(116−678)1/3+(58+378)1/33(2)2/3≈1.918. This result implies that our online algorithm is asymptotically optimal

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 reassign up to k jobs to different machines in the final assignment. For m identical machines, Albers and Hellwig (Algorithmica, 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 ~~ 1.4659. They show that k = O(m) is sufficient to achieve this bound and no 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 = Theta(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 with 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 ~~ 1.7992 with k = O(m). We also show that k = Omega(m) is necessary to achieve a competitive ratio below 2. Our algorithm is based on a subtle 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

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 ≈1.4659. They show that k=O(m) is sufficient to achieve this bound and no 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 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+δ with 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 ≈1.7992 with k=O(m). We also show that 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

SRPT Is 1.86-competitive for Completion Time Scheduling

We consider the classical problem of scheduling preemptible jobs, that ar-rive over time, on identical parallel machines. The goal is to minimize the total completion time of the jobs. In standard scheduling notation of Graham et al. [5], this problem is denoted P | rj,pmtn | j cj. A pop-ular algorithm called SRPT, which always schedules the unfinished jobs with shortest remaining processing time, is known to be 2-competitive, see Phillips et al. [12, 13]. This is also the best known competitive ratio for any online algorithm. However, it is conjectured that the competitive ra-tio of SRPT is significantly less than 2. Even breaking the barrier of 2 is considered a significant step towards the final answer of this classical online problem. We improve on this open problem by showing that SRPT is 1.86-competitive. This result is obtained using the following method, which might be of general interest: We define two dependent random variables that sum up to the difference between the cost of an SRPT schedule and the cost of an optimal schedule. Then we bound the sum of the expected values of these random variables with respect to the cost of the optimal schedule, yielding the claimed competitiveness. Furthermore, we show a lower bound of 21/19 for SRPT, improving on the previously best known 12/11 due to Lu et al. [10]

On robust online scheduling algorithms

While standard parallel machine scheduling is concerned with good assignments of jobs to machines, we aim to understand how the quality of an assignment is affected if the jobs' processing times are perturbed and therefore turn out to be longer (or shorter) than declared. We focus on online scheduling with perturbations occurring at any time, such as in railway systems when trains are late. For a variety of conditions on the severity of perturbations, we present bounds on the worst case ratio of two makespans. For the first makespan, we let the online algorithm assign jobs to machines, based on the non-perturbed processing times. We compute the makespan by replacing each job's processing time with its perturbed version while still sticking to the computed assignment. The second is an optimal offline solution for the perturbed processing times. The deviation of this ratio from the competitive ratio of the online algorithm tells us about the "price of perturbations”. We analyze this setting for Graham's algorithm, and among other bounds show a competitive ratio of 2 for perturbations decreasing the processing time of a job arbitrarily, and a competitive ratio of less than 2.5 for perturbations doubling the processing time of a job. We complement these results by providing lower bounds for any online algorithm in this setting. Finally, we propose a risk-aware online algorithm tailored for the possible bounded increase of the processing time of one job, and we show that this algorithm can be worse than Graham's algorithm in some case

Special cases of online parallel job scheduling

In this paper we consider the online scheduling of jobs, which require processing on a number of machines simultaneously. These jobs are presented to a decision maker one by one, where the next job becomes known as soon as the current job is scheduled. The objective is to minimize the makespan. For the problem with three machines we give a 2.8-competitive algorithm, improving upon the 3-competitive greedy algorithm. For the special case with arbitrary number of machines, where the jobs appear in non-increasing order of machine requirement, we give a 2.4815-competitive algorithm, improving the 2.75-competitive greedy algorithm

Competitive-Ratio Approximation Schemes for Minimizing the Makespan in the Online-List Model

We consider online scheduling on multiple machines for jobs arriving one-by-one with the objective of minimizing the makespan. For any number of identical parallel or uniformly related machines, we provide a competitive-ratio approximation scheme that computes an online algorithm whose competitive ratio is arbitrarily close to the best possible competitive ratio. We also determine this value up to any desired accuracy. This is the first application of competitive-ratio approximation schemes in the online-list model. The result proves the applicability of the concept in different online models. We expect that it fosters further research on other online problems

Online Makespan Minimization with Parallel Schedules

In online makespan minimization a sequence of jobs $\sigma = J_1,..., J_n$ has to be scheduled on $m$ identical parallel machines so as to minimize the maximum completion time of any job. We investigate the problem with an essentially new model of resource augmentation. Here, an online algorithm is allowed to build several schedules in parallel while processing $\sigma$. At the end of the scheduling process the best schedule is selected. This model can be viewed as providing an online algorithm with extra space, which is invested to maintain multiple solutions. The setting is of particular interest in parallel processing environments where each processor can maintain a single or a small set of solutions. We develop a (4/3+\eps)-competitive algorithm, for any 0<\eps\leq 1, that uses a number of 1/\eps^{O(\log (1/\eps))} schedules. We also give a (1+\eps)-competitive algorithm, for any 0<\eps\leq 1, that builds a polynomial number of (m/\eps)^{O(\log (1/\eps) / \eps)} schedules. This value depends on $m$ but is independent of the input $\sigma$. The performance guarantees are nearly best possible. We show that any algorithm that achieves a competitiveness smaller than 4/3 must construct $\Omega(m)$ schedules. Our algorithms make use of novel guessing schemes that (1) predict the optimum makespan of a job sequence $\sigma$ to within a factor of 1+\eps and (2) guess the job processing times and their frequencies in $\sigma$. In (2) we have to sparsify the universe of all guesses so as to reduce the number of schedules to a constant. The competitive ratios achieved using parallel schedules are considerably smaller than those in the standard problem without resource augmentation