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

Makespan Minimization via Posted Prices

We consider job scheduling settings, with multiple machines, where jobs arrive online and choose a machine selfishly so as to minimize their cost. Our objective is the classic makespan minimization objective, which corresponds to the completion time of the last job to complete. The incentives of the selfish jobs may lead to poor performance. To reconcile the differing objectives, we introduce posted machine prices. The selfish job seeks to minimize the sum of its completion time on the machine and the posted price for the machine. Prices may be static (i.e., set once and for all before any arrival) or dynamic (i.e., change over time), but they are determined only by the past, assuming nothing about upcoming events. Obviously, such schemes are inherently truthful. We consider the competitive ratio: the ratio between the makespan achievable by the pricing scheme and that of the optimal algorithm. We give tight bounds on the competitive ratio for both dynamic and static pricing schemes for identical, restricted, related, and unrelated machine settings. Our main result is a dynamic pricing scheme for related machines that gives a constant competitive ratio, essentially matching the competitive ratio of online algorithms for this setting. In contrast, dynamic pricing gives poor performance for unrelated machines. This lower bound also exhibits a gap between what can be achieved by pricing versus what can be achieved by online algorithms

Non-Preemptive Scheduling on Machines with Setup Times

Consider the problem in which n jobs that are classified into k types are to be scheduled on m identical machines without preemption. A machine requires a proper setup taking s time units before processing jobs of a given type. The objective is to minimize the makespan of the resulting schedule. We design and analyze an approximation algorithm that runs in time polynomial in n, m and k and computes a solution with an approximation factor that can be made arbitrarily close to 3/2.Comment: A conference version of this paper has been accepted for publication in the proceedings of the 14th Algorithms and Data Structures Symposium (WADS

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

Online Bin Covering with Limited Migration

Semi-online models where decisions may be revoked in a limited way have been studied extensively in the last years. This is motivated by the fact that the pure online model is often too restrictive to model real-world applications, where some changes might be allowed. A well-studied measure of the amount of decisions that can be revoked is the migration factor beta: When an object o of size s(o) arrives, the decisions for objects of total size at most beta * s(o) may be revoked. Usually beta should be a constant. This means that a small object only leads to small changes. This measure has been successfully investigated for different, classical problems such as bin packing or makespan minimization. The dual of makespan minimization - the Santa Claus or machine covering problem - has also been studied, whereas the dual of bin packing - the bin covering problem - has not been looked at from such a perspective. In this work, we extensively study the bin covering problem with migration in different scenarios. We develop algorithms both for the static case - where only insertions are allowed - and for the dynamic case, where items may also depart. We also develop lower bounds for these scenarios both for amortized migration and for worst-case migration showing that our algorithms have nearly optimal migration factor and asymptotic competitive ratio (up to an arbitrary small epsilon). We therefore resolve the competitiveness of the bin covering problem with migration

Scheduling parallel jobs to minimize the makespan

We consider the NP-hard problem of scheduling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job requires simultaneously a prespecified, job-dependent number of machines when being processed. We prove that the makespan of any nonpreemptive list-schedule is within a factor of 2 of the optimal preemptive makespan. This gives the best-known approximation algorithms for both the preemptive and the nonpreemptive variant of the problem. We also show that no list-scheduling algorithm can achieve a better performance guarantee than 2 for the nonpreemptive problem, no matter which priority list is chosen. List-scheduling also works in the online setting where jobs arrive over time and the length of a job becomes known only when it completes; it therefore yields a deterministic online algorithm with competitive ratio 2 as well. In addition, we consider a different online model in which jobs arrive one by one and need to be scheduled before the next job becomes known. We show that no list-scheduling algorithm has a constant competitive ratio. Still, we present the first online algorithm for scheduling parallel jobs with a constant competitive ratio in this context. We also prove a new information-theoretic lower bound of 2.25 for the competitive ratio of any deterministic online algorithm for this model. Moreover, we show that 6/5 is a lower bound for the competitive ratio of any deterministic online algorithm of the preemptive version of the model jobs arriving over tim

Scheduling with processing set restrictions : a survey

2008-2009 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe