8 research outputs found
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
Online Makespan Minimization with Parallel Schedules
In online makespan minimization a sequence of jobs
has to be scheduled on identical parallel machines so as to minimize the
maximum completion time of any job. We investigate the problem with an
essentially new model of resource augmentation. Here, an online algorithm is
allowed to build several schedules in parallel while processing . At
the end of the scheduling process the best schedule is selected. This model can
be viewed as providing an online algorithm with extra space, which is invested
to maintain multiple solutions. The setting is of particular interest in
parallel processing environments where each processor can maintain a single or
a small set of solutions.
We develop a (4/3+\eps)-competitive algorithm, for any 0<\eps\leq 1, that
uses a number of 1/\eps^{O(\log (1/\eps))} schedules. We also give a
(1+\eps)-competitive algorithm, for any 0<\eps\leq 1, that builds a
polynomial number of (m/\eps)^{O(\log (1/\eps) / \eps)} schedules. This value
depends on but is independent of the input . The performance
guarantees are nearly best possible. We show that any algorithm that achieves a
competitiveness smaller than 4/3 must construct schedules. Our
algorithms make use of novel guessing schemes that (1) predict the optimum
makespan of a job sequence to within a factor of 1+\eps and (2)
guess the job processing times and their frequencies in . In (2) we
have to sparsify the universe of all guesses so as to reduce the number of
schedules to a constant.
The competitive ratios achieved using parallel schedules are considerably
smaller than those in the standard problem without resource augmentation
Scheduling in the Random-Order Model
Makespan minimization on identical machines is a fundamental problem in online scheduling. The goal is to assign a sequence of jobs to m identical parallel machines so as to minimize the maximum completion time of any job. Already in the 1960s, Graham showed that Greedy is (2-1/m)-competitive [Graham, 1966]. The best deterministic online algorithm currently known achieves a competitive ratio of 1.9201 [Fleischer and Wahl, 2000]. No deterministic online strategy can obtain a competitiveness smaller than 1.88 [Rudin III, 2001].
In this paper, we study online makespan minimization in the popular random-order model, where the jobs of a given input arrive as a random permutation. It is known that Greedy does not attain a competitive factor asymptotically smaller than 2 in this setting [Osborn and Torng, 2008]. We present the first improved performance guarantees. Specifically, we develop a deterministic online algorithm that achieves a competitive ratio of 1.8478. The result relies on a new analysis approach. We identify a set of properties that a random permutation of the input jobs satisfies with high probability. Then we conduct a worst-case analysis of our algorithm, for the respective class of permutations. The analysis implies that the stated competitiveness holds not only in expectation but with high probability. Moreover, it provides mathematical evidence that job sequences leading to higher performance ratios are extremely rare, pathological inputs. We complement the results by lower bounds for the random-order model. We show that no deterministic online algorithm can achieve a competitive ratio smaller than 4/3. Moreover, no deterministic online algorithm can attain a competitiveness smaller than 3/2 with high probability
Scheduling in the Secretary Model
This paper studies online makespan minimization in the secretary model. Jobs, specified by their processing times, are presented in a uniformly random order. The input size n is known in advance. An online algorithm has to non-preemptively assign each job permanently and irrevocably to one of m parallel and identical machines such that the expected time it takes to process them all, the makespan, is minimized.
We give two deterministic algorithms. First, a straightforward adaptation of the semi-online strategy Light Load [Albers and Hellwig, 2012] provides a very simple approach retaining its competitive ratio of 1.75. A new and sophisticated algorithm is 1.535-competitive. These competitive ratios are not only obtained in expectation but, in fact, for all but a very tiny fraction of job orders.
Classically, online makespan minimization only considers the worst-case order. Here, no competitive ratio below 1.885 for deterministic algorithms and 1.581 using randomization is possible. The best randomized algorithm so far is 1.916-competitive. Our results show that classical worst-case orders are quite rare and pessimistic for many applications.
We complement our results by providing first lower bounds. A competitive ratio obtained on nearly all possible job orders must be at least 1.257. This implies a lower bound of 1.043 for both deterministic and randomized algorithms in the general model
On the Value of Job Migration in Online Makespan Minimization
Makespan minimization on identical parallel machines is a classical
scheduling problem. We consider the online scenario where a sequence of
jobs has to be scheduled non-preemptively on machines so as to minimize the
maximum completion time of any job. The best competitive ratio that can be
achieved by deterministic online algorithms is in the range .
Currently no randomized online algorithm with a smaller competitiveness is
known, for general .
In this paper we explore the power of job migration, i.e.\ an online
scheduler is allowed to perform a limited number of job reassignments.
Migration is a common technique used in theory and practice to balance load in
parallel processing environments. As our main result we settle the performance
that can be achieved by deterministic online algorithms. We develop an
algorithm that is -competitive, for any , where
is the solution of a certain equation. For , and
. Here is the lower branch of the Lambert function.
For , the algorithm uses at most migration operations. For
smaller , to operations may be performed. We complement this
result by a matching lower bound: No online algorithm that uses job
migrations can achieve a competitive ratio smaller than . We finally
trade performance for migrations. We give a family of algorithms that is
-competitive, for any . For , the strategy uses at
most job migrations. For , at most migrations are used.Comment: Revised versio
Online makespan scheduling with job migration on uniform machines
In the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and ≈1.4659. They show that k=O(m) is sufficient to achieve this bound and no k=o(n) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a δ=Θ(1) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than 1.4659+δ with k=o(n). We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and ≈1.7992 with k=O(m). We also show that k=Ω(m) is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines