62,928 research outputs found
Renting a Cloud
We consider the problem of efficiently scheduling jobs on data centers to minimize the cost of renting machines from "the cloud."
In the most basic cloud service model, cloud providers offer computers on demand from large pools installed in data centers.
Clients pay for use at an hourly rate. In order to minimize cost, each client needs to decide on the number of machines to be rented and the duration of renting each machine. This suggests the following
optimization problem, which we call Rent Minimization. There is a set J={j_1,j_2,...,j_n} of n jobs. Job j_i is released at time
r_i >= 0, has a deadline of d_i, and requires p_i>0 contiguous processing time, r_i,d_i,p_i in R. The jobs need to be scheduled on
identical parallel machines. Machines may be rented for any length of time; however, the cost of renting a machine for l>=0 time units is [l/D] (the smallest integer >= l/D) dollars, for some given large real D; in particular, one pays dollar 2 whether the machine is rented for D+1 or 2D time units. The goal is to schedule all the jobs in a way that minimizes the incurred rental cost.
In this paper, we develop offline and online algorithms for Rent Minimization problem. The algorithms achieve a constant factor approximation for the offline version and O(log(p_max/p_min)) for the online version, where p_max and p_min are the maximum and minimum processing time of the jobs respectively. We also show that no deterministic online algorithm can achieve an approximation factor better than log_{3}(p_max/p_min) within a constant factor. Both of these algorithms use the well-studied problem of Machine Minimization as a subroutine. Machine Minimization is a special case of Rent Minimization where D = max_{i}{d_i}. In the process of solving the Rent Minimization problem, in this paper, we also develop the first online algorithm for Machine Minimization
Makespan Minimization via Posted Prices
We consider job scheduling settings, with multiple machines, where jobs
arrive online and choose a machine selfishly so as to minimize their cost. Our
objective is the classic makespan minimization objective, which corresponds to
the completion time of the last job to complete. The incentives of the selfish
jobs may lead to poor performance. To reconcile the differing objectives, we
introduce posted machine prices. The selfish job seeks to minimize the sum of
its completion time on the machine and the posted price for the machine. Prices
may be static (i.e., set once and for all before any arrival) or dynamic (i.e.,
change over time), but they are determined only by the past, assuming nothing
about upcoming events. Obviously, such schemes are inherently truthful.
We consider the competitive ratio: the ratio between the makespan achievable
by the pricing scheme and that of the optimal algorithm. We give tight bounds
on the competitive ratio for both dynamic and static pricing schemes for
identical, restricted, related, and unrelated machine settings. Our main result
is a dynamic pricing scheme for related machines that gives a constant
competitive ratio, essentially matching the competitive ratio of online
algorithms for this setting. In contrast, dynamic pricing gives poor
performance for unrelated machines. This lower bound also exhibits a gap
between what can be achieved by pricing versus what can be achieved by online
algorithms
A General Framework for Learning-Augmented Online Allocation
Online allocation is a broad class of problems where items arriving online have to be allocated to agents who have a fixed utility/cost for each assigned item so to maximize/minimize some objective. This framework captures a broad range of fundamental problems such as the Santa Claus problem (maximizing minimum utility), Nash welfare maximization (maximizing geometric mean of utilities), makespan minimization (minimizing maximum cost), minimization of ?_p-norms, and so on. We focus on divisible items (i.e., fractional allocations) in this paper. Even for divisible items, these problems are characterized by strong super-constant lower bounds in the classical worst-case online model.
In this paper, we study online allocations in the learning-augmented setting, i.e., where the algorithm has access to some additional (machine-learned) information about the problem instance. We introduce a general algorithmic framework for learning-augmented online allocation that produces nearly optimal solutions for this broad range of maximization and minimization objectives using only a single learned parameter for every agent. As corollaries of our general framework, we improve prior results of Lattanzi et al. (SODA 2020) and Li and Xian (ICML 2021) for learning-augmented makespan minimization, and obtain the first learning-augmented nearly-optimal algorithms for the other objectives such as Santa Claus, Nash welfare, ?_p-minimization, etc. We also give tight bounds on the resilience of our algorithms to errors in the learned parameters, and study the learnability of these parameters
A General Framework for Learning-Augmented Online Allocation
Online allocation is a broad class of problems where items arriving online
have to be allocated to agents who have a fixed utility/cost for each assigned
item so to maximize/minimize some objective. This framework captures a broad
range of fundamental problems such as the Santa Claus problem (maximizing
minimum utility), Nash welfare maximization (maximizing geometric mean of
utilities), makespan minimization (minimizing maximum cost), minimization of
-norms, and so on. We focus on divisible items (i.e., fractional
allocations) in this paper. Even for divisible items, these problems are
characterized by strong super-constant lower bounds in the classical worst-case
online model.
In this paper, we study online allocations in the {\em learning-augmented}
setting, i.e., where the algorithm has access to some additional
(machine-learned) information about the problem instance. We introduce a {\em
general} algorithmic framework for learning-augmented online allocation that
produces nearly optimal solutions for this broad range of maximization and
minimization objectives using only a single learned parameter for every agent.
As corollaries of our general framework, we improve prior results of Lattanzi
et al. (SODA 2020) and Li and Xian (ICML 2021) for learning-augmented makespan
minimization, and obtain the first learning-augmented nearly-optimal algorithms
for the other objectives such as Santa Claus, Nash welfare,
-minimization, etc. We also give tight bounds on the resilience of our
algorithms to errors in the learned parameters, and study the learnability of
these parameters
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
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
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
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