15,429 research outputs found
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;
Energy-efficient algorithms for non-preemptive speed-scaling
We improve complexity bounds for energy-efficient speed scheduling problems
for both the single processor and multi-processor cases. Energy conservation
has become a major concern, so revisiting traditional scheduling problems to
take into account the energy consumption has been part of the agenda of the
scheduling community for the past few years.
We consider the energy minimizing speed scaling problem introduced by Yao et
al. where we wish to schedule a set of jobs, each with a release date, deadline
and work volume, on a set of identical processors. The processors may change
speed as a function of time and the energy they consume is the th power
of its speed. The objective is then to find a feasible schedule which minimizes
the total energy used.
We show that in the setting with an arbitrary number of processors where all
work volumes are equal, there is a approximation algorithm, where
is the generalized Bell number. This is the first constant
factor algorithm for this problem. This algorithm extends to general unequal
processor-dependent work volumes, up to losing a factor of
in the approximation, where is the maximum
ratio between two work volumes. We then show this latter problem is APX-hard,
even in the special case when all release dates and deadlines are equal and
is 4.
In the single processor case, we introduce a new linear programming
formulation of speed scaling and prove that its integrality gap is at most
. As a corollary, we obtain a
approximation algorithm where there is a single processor, improving on the
previous best bound of
when
Scheduling Parallel Jobs with Linear Speedup
We consider a scheduling problem where a set of jobs is distributed over parallel machines. The processing time of any job is dependent on the usage of a scarce renewable resource, e.g., personnel. An amount of k units of that resource can be allocated to the jobs at any time, and the more of that resource is allocated to a job, the smaller its processing time. The dependence of processing times on the amount of resources is linear for any job. The objective is to find a resource allocation and a schedule that minimizes the makespan. Utilizing an integer quadratic programming relaxation, we show how to obtain a (3+e)-approximation algorithm for that problem, for any e>0. This generalizes and improves previous results, respectively. Our approach relies on a fully polynomial time approximation scheme to solve the quadratic programming relaxation. This result is interesting in itself, because the underlying quadratic program is NP-hard to solve in general. We also briefly discuss variants of the problem and derive lower bounds.operations research and management science;
Optimizing egalitarian performance in the side-effects model of colocation for data center resource management
In data centers, up to dozens of tasks are colocated on a single physical
machine. Machines are used more efficiently, but tasks' performance
deteriorates, as colocated tasks compete for shared resources. As tasks are
heterogeneous, the resulting performance dependencies are complex. In our
previous work [18] we proposed a new combinatorial optimization model that uses
two parameters of a task - its size and its type - to characterize how a task
influences the performance of other tasks allocated to the same machine.
In this paper, we study the egalitarian optimization goal: maximizing the
worst-off performance. This problem generalizes the classic makespan
minimization on multiple processors (P||Cmax). We prove that
polynomially-solvable variants of multiprocessor scheduling are NP-hard and
hard to approximate when the number of types is not constant. For a constant
number of types, we propose a PTAS, a fast approximation algorithm, and a
series of heuristics. We simulate the algorithms on instances derived from a
trace of one of Google clusters. Algorithms aware of jobs' types lead to better
performance compared with algorithms solving P||Cmax.
The notion of type enables us to model degeneration of performance caused by
using standard combinatorial optimization methods. Types add a layer of
additional complexity. However, our results - approximation algorithms and good
average-case performance - show that types can be handled efficiently.Comment: Author's version of a paper published in Euro-Par 2017 Proceedings,
extends the published paper with addtional results and proof
How the structure of precedence constraints may change the complexity class of scheduling problems
This survey aims at demonstrating that the structure of precedence
constraints plays a tremendous role on the complexity of scheduling problems.
Indeed many problems can be NP-hard when considering general precedence
constraints, while they become polynomially solvable for particular precedence
constraints. We also show that there still are many very exciting challenges in
this research area
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