7,017 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
Energy Efficient Scheduling via Partial Shutdown
Motivated by issues of saving energy in data centers we define a collection
of new problems referred to as "machine activation" problems. The central
framework we introduce considers a collection of machines (unrelated or
related) with each machine having an {\em activation cost} of . There
is also a collection of jobs that need to be performed, and is
the processing time of job on machine . We assume that there is an
activation cost budget of -- we would like to {\em select} a subset of
the machines to activate with total cost and {\em find} a schedule
for the jobs on the machines in minimizing the makespan (or any other
metric).
For the general unrelated machine activation problem, our main results are
that if there is a schedule with makespan and activation cost then we
can obtain a schedule with makespan \makespanconstant T and activation cost
\costconstant A, for any . We also consider assignment costs for
jobs as in the generalized assignment problem, and using our framework, provide
algorithms that minimize the machine activation and the assignment cost
simultaneously. In addition, we present a greedy algorithm which only works for
the basic version and yields a makespan of and an activation cost .
For the uniformly related parallel machine scheduling problem, we develop a
polynomial time approximation scheme that outputs a schedule with the property
that the activation cost of the subset of machines is at most and the
makespan is at most for any
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Tighter Bounds on the Inefficiency Ratio of Stable Equilibria in Load Balancing Games
In this paper we study the inefficiency ratio of stable equilibria in load
balancing games introduced by Asadpour and Saberi [3]. We prove tighter lower
and upper bounds of 7/6 and 4/3, respectively. This improves over the best
known bounds in problem (19/18 and 3/2, respectively). Equivalently, the
results apply to the question of how well the optimum for the -norm can
approximate the -norm (makespan) in identical machines scheduling
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