4,602 research outputs found
An EPTAS for Scheduling on Unrelated Machines of Few Different Types
In the classical problem of scheduling on unrelated parallel machines, a set
of jobs has to be assigned to a set of machines. The jobs have a processing
time depending on the machine and the goal is to minimize the makespan, that is
the maximum machine load. It is well known that this problem is NP-hard and
does not allow polynomial time approximation algorithms with approximation
guarantees smaller than unless PNP. We consider the case that there
are only a constant number of machine types. Two machines have the same
type if all jobs have the same processing time for them. This variant of the
problem is strongly NP-hard already for . We present an efficient
polynomial time approximation scheme (EPTAS) for the problem, that is, for any
an assignment with makespan of length at most
times the optimum can be found in polynomial time in the
input length and the exponent is independent of . In particular
we achieve a running time of , where
denotes the input length. Furthermore, we study three other problem
variants and present an EPTAS for each of them: The Santa Claus problem, where
the minimum machine load has to be maximized; the case of scheduling on
unrelated parallel machines with a constant number of uniform types, where
machines of the same type behave like uniformly related machines; and the
multidimensional vector scheduling variant of the problem where both the
dimension and the number of machine types are constant. For the Santa Claus
problem we achieve the same running time. The results are achieved, using mixed
integer linear programming and rounding techniques
New bounds for truthful scheduling on two unrelated selfish machines
We consider the minimum makespan problem for tasks and two unrelated
parallel selfish machines. Let be the best approximation ratio of
randomized monotone scale-free algorithms. This class contains the most
efficient algorithms known for truthful scheduling on two machines. We propose
a new formulation for , as well as upper and lower bounds on
based on this formulation. For the lower bound, we exploit pointwise
approximations of cumulative distribution functions (CDFs). For the upper
bound, we construct randomized algorithms using distributions with piecewise
rational CDFs. Our method improves upon the existing bounds on for small
. In particular, we obtain almost tight bounds for showing that
.Comment: 28 pages, 3 tables, 1 figure. Theory Comput Syst (2019
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 PTAS for Minimizing Average Weighted Completion Time With Release Dates on Uniformly Related Machines
A classical scheduling problem is to find schedules that minimize average weighted completion time of jobs with release dates. When multiple machines are available, the machine environments may range from identical machines (the processing time required by a job is invariant across the machines) at one end, to unrelated machines (the processing time required by a job on any machine is an arbitrary function of the specific machine) at the other end of the spectrum. While the problem is strongly NP-hard even in the case of a single machine, constant factor approximation algorithms have been known for even the most general machine environment of unrelated machines. Recently, a polynomial-time approximation scheme (PTAS) was discovered for the case of identical parallel machines [1]. In contrast, it is known that this problem is MAX SNP-hard for unrelated machines [10]. An important open problem is to determine the approximability of the intermediate case of uniformly related machines where each machine i has a speed si and it takes p/si time to executing a job of processing size pIn this paper, we resolve this problem by obtaining a PTAS for the problem. This improves the earlier known ratio of (2 + â) for the problem
Resource assignment algorithms for vehicular clouds
In this thesis, we study the task scheduling problem in vehicular clouds. It falls in the category of unrelated parallel machine scheduling problems. Resource assignment in vehicular clouds must deal with the transient nature of the cloud resources and a relaxed definition of non-preemptive tasks. Despite a rich literature in machine scheduling and grid computing, the resource assignment problem in vehicular clouds has not been examined yet. We show that even the problem of finding a minimum cost schedule for a single task over unrelated machines is NP-hard. We then provide a fully polynomial time approximation scheme and a greedy approximation for scheduling a single task. We extend these algorithms to the case of scheduling n tasks. We validate our algorithms through extensive simulations that use synthetically generated data as well as real data extracted from vehicle mobility and grid computing workload traces. Our contributions are, to the best of our knowledge, the first quantitative analysis of the computational power of vehicular clouds
Semidefinite Relaxations for Parallel Machine Scheduling
We consider the problem of scheduling unrelated parallel machines so as to minimize the total weighted completion time of jobs.Whereas the best previously known approximation algorithms for this problem are based on LP relaxations, we give a 32âapproximation algorithm that relies on a convex quadratic programming relaxation. For the special case of two machines we present a further improvement to a 1.2752âapproximation; we introduce a more sophisticated semidefinite programming relaxation and apply the random hyperplane technique introduced by Goemans and Williams on for the MAXCUT problem and its refined version of Feige and Goemans. To the best of our knowledge, this is the first time that convex and semidefinite programming techniques (apart from LPs) are used in the area of scheduling
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
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