226 research outputs found
Online Bin Covering with Limited Migration
Semi-online models where decisions may be revoked in a limited way have been studied extensively in the last years.
This is motivated by the fact that the pure online model is often too restrictive to model real-world applications, where some changes might be allowed. A well-studied measure of the amount of decisions that can be revoked is the migration factor beta: When an object o of size s(o) arrives, the decisions for objects of total size at most beta * s(o) may be revoked. Usually beta should be a constant. This means that a small object only leads to small changes. This measure has been successfully investigated for different, classical problems such as bin packing or makespan minimization. The dual of makespan minimization - the Santa Claus or machine covering problem - has also been studied, whereas the dual of bin packing - the bin covering problem - has not been looked at from such a perspective.
In this work, we extensively study the bin covering problem with migration in different scenarios. We develop algorithms both for the static case - where only insertions are allowed - and for the dynamic case, where items may also depart. We also develop lower bounds for these scenarios both for amortized migration and for worst-case migration showing that our algorithms have nearly optimal migration factor and asymptotic competitive ratio (up to an arbitrary small epsilon). We therefore resolve the competitiveness of the bin covering problem with migration
Symmetry Exploitation for Online Machine Covering with Bounded Migration
Online models that allow recourse are highly effective in situations where classical models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this setting, jobs arrive one by one and must be assigned to machines with the objective of maximizing the minimum machine load. When a job arrives, we are allowed to reassign some jobs as long as their total size is (at most) proportional to the processing time of the arriving job. The proportionality constant is called the migration factor of the algorithm.
By rounding the processing times, which yields useful structural properties for online packing and covering problems, we design first a simple (1.7 + epsilon)-competitive algorithm using a migration factor of O(1/epsilon) which maintains at every arrival a locally optimal solution with respect to the Jump neighborhood. After that, we present as our main contribution a more involved (4/3+epsilon)-competitive algorithm using a migration factor of O~(1/epsilon^3). At every arrival, we run an adaptation of the Largest Processing Time first (LPT) algorithm. Since the new job can cause a complete change of the assignment of smaller jobs in both cases, a low migration factor is achieved by carefully exploiting the highly symmetric structure obtained by the rounding procedure
Cardinality Constrained Scheduling in Online Models
Makespan minimization on parallel identical machines is a classical and
intensively studied problem in scheduling, and a classic example for online
algorithm analysis with Graham's famous list scheduling algorithm dating back
to the 1960s. In this problem, jobs arrive over a list and upon an arrival, the
algorithm needs to assign the job to a machine. The goal is to minimize the
makespan, that is, the maximum machine load. In this paper, we consider the
variant with an additional cardinality constraint: The algorithm may assign at
most jobs to each machine where is part of the input. While the offline
(strongly NP-hard) variant of cardinality constrained scheduling is well
understood and an EPTAS exists here, no non-trivial results are known for the
online variant. We fill this gap by making a comprehensive study of various
different online models. First, we show that there is a constant competitive
algorithm for the problem and further, present a lower bound of on the
competitive ratio of any online algorithm. Motivated by the lower bound, we
consider a semi-online variant where upon arrival of a job of size , we are
allowed to migrate jobs of total size at most a constant times . This
constant is called the migration factor of the algorithm. Algorithms with small
migration factors are a common approach to bridge the performance of online
algorithms and offline algorithms. One can obtain algorithms with a constant
migration factor by rounding the size of each incoming job and then applying an
ordinal algorithm to the resulting rounded instance. With this in mind, we also
consider the framework of ordinal algorithms and characterize the competitive
ratio that can be achieved using the aforementioned approaches.Comment: An extended abstract will appear in the proceedings of STACS'2
Online load balancing with general reassignment cost
We investigate a semi-online variant of load balancing with restricted assignment. In this problem, we are given n jobs, which need to be processed by m machines with the goal to minimize the maximum machine load. Since strong lower bounds rule out any competitive ratio of o(logn), we may reassign jobs at a certain job-individual cost. We generalize a result by Gupta, Kumar, and Stein (SODA 2014) by giving a O(loglogmn)-competitive algorithm with constant amortized reassignment cost
Machine Covering in the Random-Order Model
In the Online Machine Covering problem jobs, defined by their sizes, arrive
one by one and have to be assigned to parallel and identical machines, with
the goal of maximizing the load of the least-loaded machine. In this work, we
study the Machine Covering problem in the recently popular random-order model.
Here no extra resources are present, but instead the adversary is weakened in
that it can only decide upon the input set while jobs are revealed uniformly at
random. It is particularly relevant to Machine Covering where lower bounds are
usually associated to highly structured input sequences.
We first analyze Graham's Greedy-strategy in this context and establish that
its competitive ratio decreases slightly to
which is asymptotically tight. Then, as
our main result, we present an improved -competitive
algorithm for the problem. This result is achieved by exploiting the extra
information coming from the random order of the jobs, using sampling techniques
to devise an improved mechanism to distinguish jobs that are relatively large
from small ones. We complement this result with a first lower bound showing
that no algorithm can have a competitive ratio of
in the random-order model. This
lower bound is achieved by studying a novel variant of the Secretary problem,
which could be of independent interest
Online Strip Packing with Polynomial Migration
We consider the relaxed online strip packing problem, where rectangular items arrive online and have to be packed into a strip of fixed width such that the packing height is minimized. Thereby, repacking of previously packed items is allowed. The amount of repacking is measured by the migration factor, defined as the total size of repacked items divided by the size of the arriving item. First, we show that no algorithm with constant migration factor can produce solutions with asymptotic ratio better than 4/3. Against this background, we allow amortized migration, i.e. to save migration for a later time step. As a main result, we present an AFPTAS with asymptotic ratio 1 + O(epsilon) for any epsilon > 0 and amortized migration factor polynomial in 1/epsilon. To our best knowledge, this is the first algorithm for online strip packing considered in a repacking model
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