2,470 research outputs found
Structural Properties of an Open Problem in Preemptive Scheduling
Structural properties of optimal preemptive schedules have been studied in a
number of recent papers with a primary focus on two structural parameters: the
minimum number of preemptions necessary, and a tight lower bound on `shifts',
i.e., the sizes of intervals bounded by the times created by preemptions, job
starts, or completions. So far only rough bounds for these parameters have been
derived for specific problems. This paper sharpens the bounds on these
structural parameters for a well-known open problem in the theory of preemptive
scheduling: Instances consist of in-trees of unit-execution-time jobs with
release dates, and the objective is to minimize the total completion time on
two processors. This is among the current, tantalizing `threshold' problems of
scheduling theory: Our literature survey reveals that any significant
generalization leads to an NP-hard problem, but that any significant
simplification leads to tractable problem.
For the above problem, we show that the number of preemptions necessary for
optimality need not exceed ; that the number must be of order
for some instances; and that the minimum shift need not be
less than . These bounds are obtained by combinatorial analysis of
optimal schedules rather than by the analysis of polytope corners for
linear-program formulations, an approach to be found in earlier papers. The
bounds immediately follow from a fundamental structural property called
`normality', by which minimal shifts of a job are exponentially decreasing
functions. In particular, the first interval between a preempted job's start
and its preemption is a multiple of 1/2, the second such interval is a multiple
of 1/4, and in general, the -th preemption occurs at a multiple of .
We expect the new structural properties to play a prominent role in finally
settling a vexing, still-open question of complexity
On the periodic behavior of real-time schedulers on identical multiprocessor platforms
This paper is proposing a general periodicity result concerning any
deterministic and memoryless scheduling algorithm (including
non-work-conserving algorithms), for any context, on identical multiprocessor
platforms. By context we mean the hardware architecture (uniprocessor,
multicore), as well as task constraints like critical sections, precedence
constraints, self-suspension, etc. Since the result is based only on the
releases and deadlines, it is independent from any other parameter. Note that
we do not claim that the given interval is minimal, but it is an upper bound
for any cycle of any feasible schedule provided by any deterministic and
memoryless scheduler
Preemptive Multi-Machine Scheduling of Equal-Length Jobs to Minimize the Average Flow Time
We study the problem of preemptive scheduling of n equal-length jobs with
given release times on m identical parallel machines. The objective is to
minimize the average flow time. Recently, Brucker and Kravchenko proved that
the optimal schedule can be computed in polynomial time by solving a linear
program with O(n^3) variables and constraints, followed by some substantial
post-processing (where n is the number of jobs.) In this note we describe a
simple linear program with only O(mn) variables and constraints. Our linear
program produces directly the optimal schedule and does not require any
post-processing
The Complexity of Mean Flow Time Scheduling Problems with Release Times
We study the problem of preemptive scheduling n jobs with given release times
on m identical parallel machines. The objective is to minimize the average flow
time. We show that when all jobs have equal processing times then the problem
can be solved in polynomial time using linear programming. Our algorithm can
also be applied to the open-shop problem with release times and unit processing
times. For the general case (when processing times are arbitrary), we show that
the problem is unary NP-hard.Comment: Subsumes and replaces cs.DS/0412094 and "Complexity of mean flow time
scheduling problems with release dates" by P.B, S.
Optimal Algorithms for Scheduling under Time-of-Use Tariffs
We consider a natural generalization of classical scheduling problems in
which using a time unit for processing a job causes some time-dependent cost
which must be paid in addition to the standard scheduling cost. We study the
scheduling objectives of minimizing the makespan and the sum of (weighted)
completion times. It is not difficult to derive a polynomial-time algorithm for
preemptive scheduling to minimize the makespan on unrelated machines. The
problem of minimizing the total (weighted) completion time is considerably
harder, even on a single machine. We present a polynomial-time algorithm that
computes for any given sequence of jobs an optimal schedule, i.e., the optimal
set of time-slots to be used for scheduling jobs according to the given
sequence. This result is based on dynamic programming using a subtle analysis
of the structure of optimal solutions and a potential function argument. With
this algorithm, we solve the unweighted problem optimally in polynomial time.
For the more general problem, in which jobs may have individual weights, we
develop a polynomial-time approximation scheme (PTAS) based on a dual
scheduling approach introduced for scheduling on a machine of varying speed. As
the weighted problem is strongly NP-hard, our PTAS is the best possible
approximation we can hope for.Comment: 17 pages; A preliminary version of this paper with a subset of
results appeared in the Proceedings of MFCS 201
Better Unrelated Machine Scheduling for Weighted Completion Time via Random Offsets from Non-Uniform Distributions
In this paper we consider the classic scheduling problem of minimizing total
weighted completion time on unrelated machines when jobs have release times,
i.e, using the three-field notation. For this
problem, a 2-approximation is known based on a novel convex programming (J. ACM
2001 by Skutella). It has been a long standing open problem if one can improve
upon this 2-approximation (Open Problem 8 in J. of Sched. 1999 by Schuurman and
Woeginger). We answer this question in the affirmative by giving a
1.8786-approximation. We achieve this via a surprisingly simple linear
programming, but a novel rounding algorithm and analysis. A key ingredient of
our algorithm is the use of random offsets sampled from non-uniform
distributions.
We also consider the preemptive version of the problem, i.e, . We again use the idea of sampling offsets from non-uniform
distributions to give the first better than 2-approximation for this problem.
This improvement also requires use of a configuration LP with variables for
each job's complete schedules along with more careful analysis. For both
non-preemptive and preemptive versions, we break the approximation barrier of 2
for the first time.Comment: 24 pages. To apper in FOCS 201
Competitive-Ratio Approximation Schemes for Minimizing the Makespan in the Online-List Model
We consider online scheduling on multiple machines for jobs arriving
one-by-one with the objective of minimizing the makespan. For any number of
identical parallel or uniformly related machines, we provide a
competitive-ratio approximation scheme that computes an online algorithm whose
competitive ratio is arbitrarily close to the best possible competitive ratio.
We also determine this value up to any desired accuracy. This is the first
application of competitive-ratio approximation schemes in the online-list
model. The result proves the applicability of the concept in different online
models. We expect that it fosters further research on other online problems
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