2,802 research outputs found
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
Throughput Maximization in the Speed-Scaling Setting
We are given a set of jobs and a single processor that can vary its speed
dynamically. Each job is characterized by its processing requirement
(work) , its release date and its deadline . We are also given
a budget of energy and we study the scheduling problem of maximizing the
throughput (i.e. the number of jobs which are completed on time). We propose a
dynamic programming algorithm that solves the preemptive case of the problem,
i.e. when the execution of the jobs may be interrupted and resumed later, in
pseudo-polynomial time. Our algorithm can be adapted for solving the weighted
version of the problem where every job is associated with a weight and
the objective is the maximization of the sum of the weights of the jobs that
are completed on time. Moreover, we provide a strongly polynomial time
algorithm to solve the non-preemptive unweighed case when the jobs have the
same processing requirements. For the weighted case, our algorithm can be
adapted for solving the non-preemptive version of the problem in
pseudo-polynomial time.Comment: submitted to SODA 201
Approximation Algorithms for Correlated Knapsacks and Non-Martingale Bandits
In the stochastic knapsack problem, we are given a knapsack of size B, and a
set of jobs whose sizes and rewards are drawn from a known probability
distribution. However, we know the actual size and reward only when the job
completes. How should we schedule jobs to maximize the expected total reward?
We know O(1)-approximations when we assume that (i) rewards and sizes are
independent random variables, and (ii) we cannot prematurely cancel jobs. What
can we say when either or both of these assumptions are changed?
The stochastic knapsack problem is of interest in its own right, but
techniques developed for it are applicable to other stochastic packing
problems. Indeed, ideas for this problem have been useful for budgeted learning
problems, where one is given several arms which evolve in a specified
stochastic fashion with each pull, and the goal is to pull the arms a total of
B times to maximize the reward obtained. Much recent work on this problem focus
on the case when the evolution of the arms follows a martingale, i.e., when the
expected reward from the future is the same as the reward at the current state.
What can we say when the rewards do not form a martingale?
In this paper, we give constant-factor approximation algorithms for the
stochastic knapsack problem with correlations and/or cancellations, and also
for budgeted learning problems where the martingale condition is not satisfied.
Indeed, we can show that previously proposed LP relaxations have large
integrality gaps. We propose new time-indexed LP relaxations, and convert the
fractional solutions into distributions over strategies, and then use the LP
values and the time ordering information from these strategies to devise a
randomized adaptive scheduling algorithm. We hope our LP formulation and
decomposition methods may provide a new way to address other correlated bandit
problems with more general contexts
Throughput Maximization in Multiprocessor Speed-Scaling
We are given a set of jobs that have to be executed on a set of
speed-scalable machines that can vary their speeds dynamically using the energy
model introduced in [Yao et al., FOCS'95]. Every job is characterized by
its release date , its deadline , its processing volume if
is executed on machine and its weight . We are also given a budget
of energy and our objective is to maximize the weighted throughput, i.e.
the total weight of jobs that are completed between their respective release
dates and deadlines. We propose a polynomial-time approximation algorithm where
the preemption of the jobs is allowed but not their migration. Our algorithm
uses a primal-dual approach on a linearized version of a convex program with
linear constraints. Furthermore, we present two optimal algorithms for the
non-preemptive case where the number of machines is bounded by a fixed
constant. More specifically, we consider: {\em (a)} the case of identical
processing volumes, i.e. for every and , for which we
present a polynomial-time algorithm for the unweighted version, which becomes a
pseudopolynomial-time algorithm for the weighted throughput version, and {\em
(b)} the case of agreeable instances, i.e. for which if and only
if , for which we present a pseudopolynomial-time algorithm. Both
algorithms are based on a discretization of the problem and the use of dynamic
programming
Some complexity and approximation results for coupled-tasks scheduling problem according to topology
We consider the makespan minimization coupled-tasks problem in presence of
compatibility constraints with a specified topology. In particular, we focus on
stretched coupled-tasks, i.e. coupled-tasks having the same sub-tasks execution
time and idle time duration. We study several problems in framework of classic
complexity and approximation for which the compatibility graph is bipartite
(star, chain,. . .). In such a context, we design some efficient
polynomial-time approximation algorithms for an intractable scheduling problem
according to some parameters
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