1,139 research outputs found
Online Makespan Minimization with Parallel Schedules
In online makespan minimization a sequence of jobs
has to be scheduled on 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 . 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 but is independent of the input . The performance
guarantees are nearly best possible. We show that any algorithm that achieves a
competitiveness smaller than 4/3 must construct schedules. Our
algorithms make use of novel guessing schemes that (1) predict the optimum
makespan of a job sequence to within a factor of 1+\eps and (2)
guess the job processing times and their frequencies in . 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
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
Energy-Efficient Multiprocessor Scheduling for Flow Time and Makespan
We consider energy-efficient scheduling on multiprocessors, where the speed
of each processor can be individually scaled, and a processor consumes power
when running at speed , for . A scheduling algorithm
needs to decide at any time both processor allocations and processor speeds for
a set of parallel jobs with time-varying parallelism. The objective is to
minimize the sum of the total energy consumption and certain performance
metric, which in this paper includes total flow time and makespan. For both
objectives, we present instantaneous parallelism clairvoyant (IP-clairvoyant)
algorithms that are aware of the instantaneous parallelism of the jobs at any
time but not their future characteristics, such as remaining parallelism and
work. For total flow time plus energy, we present an -competitive
algorithm, which significantly improves upon the best known non-clairvoyant
algorithm and is the first constant competitive result on multiprocessor speed
scaling for parallel jobs. In the case of makespan plus energy, which is
considered for the first time in the literature, we present an
-competitive algorithm, where is the total number of
processors. We show that this algorithm is asymptotically optimal by providing
a matching lower bound. In addition, we also study non-clairvoyant scheduling
for total flow time plus energy, and present an algorithm that achieves -competitive for jobs with arbitrary release time and
-competitive for jobs with identical release time. Finally,
we prove an lower bound on the competitive ratio of
any non-clairvoyant algorithm, matching the upper bound of our algorithm for
jobs with identical release time
Inapproximability Results for Scheduling with Interval and Resource Restrictions
In the restricted assignment problem, the input consists of a set of machines and a set of jobs each with a processing time and a subset of eligible machines. The goal is to find an assignment of the jobs to the machines minimizing the makespan, that is, the maximum summed up processing time any machine receives. Herein, jobs should only be assigned to those machines on which they are eligible. It is well-known that there is no polynomial time approximation algorithm with an approximation guarantee of less than 1.5 for the restricted assignment problem unless P=NP. In this work, we show hardness results for variants of the restricted assignment problem with particular types of restrictions.
For the case of interval restrictions - where the machines can be totally ordered such that jobs are eligible on consecutive machines - we show that there is no polynomial time approximation scheme (PTAS) unless P=NP. The question of whether a PTAS for this variant exists was stated as an open problem before, and PTAS results for special cases of this variant are known.
Furthermore, we consider a variant with resource restriction where the sets of eligible machines are of the following form: There is a fixed number of (renewable) resources, each machine has a capacity, and each job a demand for each resource. A job is eligible on a machine if its demand is at most as big as the capacity of the machine for each resource. For one resource, this problem has been intensively studied under several different names and is known to admit a PTAS, and for two resources the variant with interval restrictions is contained as a special case. Moreover, the version with multiple resources is closely related to makespan minimization on parallel machines with a low rank processing time matrix. We show that there is no polynomial time approximation algorithm with a rate smaller than 48/47 ? 1.02 or 1.5 for scheduling with resource restrictions with 2 or 4 resources, respectively, unless P=NP. All our results can be extended to the so called Santa Claus variants of the problems where the goal is to maximize the minimal processing time any machine receives
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
Games and Mechanism Design in Machine Scheduling – An Introduction
In this paper, we survey different models, techniques, and some recent results to tackle machine scheduling problems within a distributed setting. In traditional optimization, a central authority is asked to solve a (computationally hard) optimization problem. In contrast, in distributed settings there are several agents, possibly equipped with private information that is not publicly known, and these agents need to interact in order to derive a solution to the problem. Usually the agents have their individual preferences, which induces them to behave strategically in order to manipulate the resulting solution. Nevertheless, one is often interested in the global performance of such systems. The analysis of such distributed settings requires techniques from classical Optimization, Game Theory, and Economic Theory. The paper therefore briefly introduces the most important of the underlying concepts, and gives a selection of typical research questions and recent results, focussing on applications to machine scheduling problems. This includes the study of the so-called price of anarchy for settings where the agents do not possess private information, as well as the design and analysis of (truthful) mechanisms in settings where the agents do possess private information.computer science applications;
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