11,721 research outputs found
New Results on Online Resource Minimization
We consider the online resource minimization problem in which jobs with hard
deadlines arrive online over time at their release dates. The task is to
determine a feasible schedule on a minimum number of machines. We rigorously
study this problem and derive various algorithms with small constant
competitive ratios for interesting restricted problem variants. As the most
important special case, we consider scheduling jobs with agreeable deadlines.
We provide the first constant ratio competitive algorithm for the
non-preemptive setting, which is of particular interest with regard to the
known strong lower bound of n for the general problem. For the preemptive
setting, we show that the natural algorithm LLF achieves a constant ratio for
agreeable jobs, while for general jobs it has a lower bound of Omega(n^(1/3)).
We also give an O(log n)-competitive algorithm for the general preemptive
problem, which improves upon the known O(p_max/p_min)-competitive algorithm.
Our algorithm maintains a dynamic partition of the job set into loose and tight
jobs and schedules each (temporal) subset individually on separate sets of
machines. The key is a characterization of how the decrease in the relative
laxity of jobs influences the optimum number of machines. To achieve this we
derive a compact expression of the optimum value, which might be of independent
interest. We complement the general algorithmic result by showing lower bounds
that rule out that other known algorithms may yield a similar performance
guarantee
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
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
Polynomiality for Bin Packing with a Constant Number of Item Types
We consider the bin packing problem with d different item sizes s_i and item
multiplicities a_i, where all numbers are given in binary encoding. This
problem formulation is also known as the 1-dimensional cutting stock problem.
In this work, we provide an algorithm which, for constant d, solves bin
packing in polynomial time. This was an open problem for all d >= 3.
In fact, for constant d our algorithm solves the following problem in
polynomial time: given two d-dimensional polytopes P and Q, find the smallest
number of integer points in P whose sum lies in Q.
Our approach also applies to high multiplicity scheduling problems in which
the number of copies of each job type is given in binary encoding and each type
comes with certain parameters such as release dates, processing times and
deadlines. We show that a variety of high multiplicity scheduling problems can
be solved in polynomial time if the number of job types is constant
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