612 research outputs found
Malleable Scheduling Beyond Identical Machines
In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. Jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S for j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than e/(e-1), unless P = NP. On the positive side, we present polynomial-time algorithms with approximation ratios 2e/(e-1) for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding and result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of 1+phi for unrelated speeds (phi is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms (i) for minimizing the sum of weighted completion times; and (ii) a variant where we determine the effective speed of a set of allocated machines based on the L_p norm of their speeds
Efficient Algorithms for Scheduling Moldable Tasks
We study the problem of scheduling independent moldable tasks on
processors that arises in large-scale parallel computations. When tasks are
monotonic, the best known result is a -approximation
algorithm for makespan minimization with a complexity linear in and
polynomial in and where is
arbitrarily small. We propose a new perspective of the existing speedup models:
the speedup of a task is linear when the number of assigned
processors is small (up to a threshold ) while it presents
monotonicity when ranges in ; the bound
indicates an unacceptable overhead when parallelizing on too many processors.
For a given integer , let . In this paper, we propose a -approximation algorithm for makespan minimization with a
complexity where
(). As
a by-product, we also propose a -approximation algorithm for
throughput maximization with a common deadline with a complexity
Scheduling Monotone Moldable Jobs in Linear Time
A moldable job is a job that can be executed on an arbitrary number of
processors, and whose processing time depends on the number of processors
allotted to it. A moldable job is monotone if its work doesn't decrease for an
increasing number of allotted processors. We consider the problem of scheduling
monotone moldable jobs to minimize the makespan.
We argue that for certain compact input encodings a polynomial algorithm has
a running time polynomial in n and log(m), where n is the number of jobs and m
is the number of machines. We describe how monotony of jobs can be used to
counteract the increased problem complexity that arises from compact encodings,
and give tight bounds on the approximability of the problem with compact
encoding: it is NP-hard to solve optimally, but admits a PTAS.
The main focus of this work are efficient approximation algorithms. We
describe different techniques to exploit the monotony of the jobs for better
running times, and present a (3/2+{\epsilon})-approximate algorithm whose
running time is polynomial in log(m) and 1/{\epsilon}, and only linear in the
number n of jobs
Split Scheduling with Uniform Setup Times
We study a scheduling problem in which jobs may be split into parts, where
the parts of a split job may be processed simultaneously on more than one
machine. Each part of a job requires a setup time, however, on the machine
where the job part is processed. During setup a machine cannot process or set
up any other job. We concentrate on the basic case in which setup times are
job-, machine-, and sequence-independent. Problems of this kind were
encountered when modelling practical problems in planning disaster relief
operations. Our main algorithmic result is a polynomial-time algorithm for
minimising total completion time on two parallel identical machines. We argue
why the same problem with three machines is not an easy extension of the
two-machine case, leaving the complexity of this case as a tantalising open
problem. We give a constant-factor approximation algorithm for the general case
with any number of machines and a polynomial-time approximation scheme for a
fixed number of machines. For the version with objective minimising weighted
total completion time we prove NP-hardness. Finally, we conclude with an
overview of the state of the art for other split scheduling problems with job-,
machine-, and sequence-independent setup times
Power Strip Packing of Malleable Demands in Smart Grid
We consider a problem of supplying electricity to a set of
customers in a smart-grid framework. Each customer requires a certain amount of
electrical energy which has to be supplied during the time interval . We
assume that each demand has to be supplied without interruption, with possible
duration between and , which are given system parameters (). At each moment of time, the power of the grid is the sum of all the
consumption rates for the demands being supplied at that moment. Our goal is to
find an assignment that minimizes the {\it power peak} - maximal power over
- while satisfying all the demands. To do this first we find the lower
bound of optimal power peak. We show that the problem depends on whether or not
the pair belongs to a "good" region . If it does - then
an optimal assignment almost perfectly "fills" the rectangle with being the sum of all the energy demands - thus
achieving an optimal power peak . Conversely, if do not belong to
, we identify the lower bound on the optimal value of
power peak and introduce a simple linear time algorithm that almost perfectly
arranges all the demands in a rectangle
and show that it is asymptotically optimal
Machine Scheduling with Resource Dependent Processing Times
We consider several parallel machine scheduling settings with the objective to minimize the schedule makespan. The most general of these settings is unrelated parallel machine scheduling. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource. A given amount of that resource, e.g. workers, can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes classical machine scheduling problems, adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of integer programming formulations for relaxations of the respective problems, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham''s list scheduling, we thus prove the existence of constant factor approximation algorithms. Our performance guarantee is 6.83 for the most general case of unrelated parallel machine scheduling. We improve this bound for two special cases, namely to 5.83 whenever the jobs are assigned to machines beforehand, and to (5+e), e>0, whenever the processing times do not depend on the machine. Moreover, we discuss tightness of the relaxations, and derive inapproximability results.operations research and management science;
A (3/2+É) approximation algorithm for scheduling malleable and non-malleable parallel tasks
In this paper we study a scheduling problem with malleable and non-malleable parallel tasks on processors. A non-malleable parallel task is one that runs in parallel on a specific given number of processors. The goal is to find a non-preemptive schedule on the processors which minimizes the makespan, or the latest task completion time. The previous best result is the list scheduling algorithm with an absolute approximation ratio of . On the other hand, there does not exist an approximation algorithm for scheduling non-malleable parallel tasks with ratio smaller than , unless . In this paper we show that a schedule with length can be computed for the scheduling problem in time . Furthermore we present an approximation algorithm for scheduling malleable parallel tasks. Finally, we show how to extend our algorithms to the variant with additional release dates
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