14,814 research outputs found
On Lazy Bin Covering and Packing problems
AbstractIn this paper, we study two interesting variants of the classical bin packing problem, called Lazy Bin Covering (LBC) and Cardinality Constrained Maximum Resource Bin Packing (CCMRBP) problems. For the offline LBC problem, we first prove the approximation ratio of the First-Fit-Decreasing and First-Fit-Increasing algorithms, then present an APTAS. For the online LBC problem, we give a competitive analysis for the algorithms of Next-Fit, Worst-Fit, First-Fit, and a modified HARMONICM algorithm. The CCMRBP problem is a generalization of the Maximum Resource Bin Packing (MRBP) problem Boyar et al. (2006) [1]. For this problem, we prove that its offline version is no harder to approximate than the offline MRBP problem
On Multi-dimensional Packing Problems
We study the approximability of multi-dimensional generalizations of three classical packing problems: multiprocessor scheduling, bin packing, and the knapsack problem. Specifically, we study the vector scheduling problem, its dual problem, namely, the vector bin packing problem, and a class of packing integer programs. The vector scheduling problem is to schedule n d-dimensional tasks on m machines such that the maximum load over all dimensions and all machines is minimized. The vector bin packing problem, on the other hand, seeks to minimize the number of bins needed to schedule all n tasks such that the maximum load on any dimension accross all bins is bounded by a fixed quantity, say 1. Such problems naturally arise when scheduling tasks that have multiple resource requirements. Finally, packing integer programs capture a core problem that directly relates to both vector scheduling and vector bin packing, namely, the problem of packing a miximum number of vectors in a single bin of unit height. We obtain a variety of new algorithmic as well as inapproximability results for these three problems
Approximation Algorithms for Round-UFP and Round-SAP
We study Round-UFP and Round-SAP, two generalizations of the classical Bin Packing problem that correspond to the unsplittable flow problem on a path (UFP) and the storage allocation problem (SAP), respectively. We are given a path with capacities on its edges and a set of jobs where for each job we are given a demand and a subpath. In Round-UFP, the goal is to find a packing of all jobs into a minimum number of copies (rounds) of the given path such that for each copy, the total demand of jobs on any edge does not exceed the capacity of the respective edge. In Round-SAP, the jobs are considered to be rectangles and the goal is to find a non-overlapping packing of these rectangles into a minimum number of rounds such that all rectangles lie completely below the capacity profile of the edges.
We show that in contrast to Bin Packing, both problems do not admit an asymptotic polynomial-time approximation scheme (APTAS), even when all edge capacities are equal. However, for this setting, we obtain asymptotic (2+?)-approximations for both problems. For the general case, we obtain an O(log log n)-approximation algorithm and an O(log log 1/?)-approximation under (1+?)-resource augmentation for both problems. For the intermediate setting of the no bottleneck assumption (i.e., the maximum job demand is at most the minimum edge capacity), we obtain an absolute 12- and an asymptotic (16+?)-approximation algorithm for Round-UFP and Round-SAP, respectively
An off-line dual maximum resource bin packing model for solving the maintenance problem in the aviation industry
In the aviation industry, propeller motor engines have a lifecycle of several thousand hours of flight and the maintenance is an important part of their lifecycle. The present article considers a multi-resource, priority-based case scheduling problem, which is applied in a Romanian manufacturing company, that repairs and maintains helicopter and airplane engines at a certain quality level imposed by the aviation standards. Given a reduced budget constraint, the management’s goal is to maximize the utilization of their resources (financial, material, space, workers), by maintaining a prior known priority rule. An Off-Line Dual Maximum Resource Bin Packing model, based on a Mixed Integer Programming model is thus presented. The obtained results show an increase with approx. 25% of the Just in Time shipping of the engines to the customers and approx. 12,5% increase in the utilization of the working area
A tale of two packing problems : improved algorithms and tighter bounds for online bin packing and the geometric knapsack problem
In this thesis, we deal with two packing problems: the online bin packing and the geometric knapsack problem. In online bin packing, the aim is to pack a given number of items of different size into a minimal number of containers. The items need to be packed one by one without knowing future items. For online bin packing in one dimension, we present a new family of algorithms that constitutes the first improvement over the previously best algorithm in almost 15 years. While the algorithmic ideas are intuitive, an elaborate analysis is required to prove its competitive ratio. We also give a lower bound for the competitive ratio of this family of algorithms. For online bin packing in higher dimensions, we discuss lower bounds for the competitive ratio and show that the ideas from the one-dimensional case cannot be easily transferred to obtain better two-dimensional algorithms. In the geometric knapsack problem, one aims to pack a maximum weight subset of given rectangles into one square container. For this problem, we consider online approximation algorithms. For geometric knapsack with square items, we improve the running time of the best known PTAS and obtain an EPTAS. This shows that large running times caused by some standard techniques for geometric packing problems are not always necessary and can be improved. Finally, we show how to use resource augmentation to compute optimal solutions in EPTAS-time, thereby improving upon the known PTAS for this case.In dieser Arbeit betrachten wir zwei Packungsprobleme: Online Bin Packing und das geometrische Rucksackproblem. Bei Online Bin Packing versucht man, eine gegebene Menge an Objekten verschiedener Größe in die kleinstmögliche Anzahl an Behältern zu packen. Die Objekte müssen eins nach dem anderen gepackt werden, ohne zukünftige Objekte zu kennen. Für eindimensionales Online Bin Packing beschreiben wir einen neuen Algorithmus, der die erste Verbesserung gegenüber dem bisher besten Algorithmus seit fast 15 Jahren darstellt. Während die algorithmischen Ideen intuitiv sind, ist eine ausgefeilte Analyse notwendig um das Kompetitivitätsverhältnis zu beweisen. Für Online Bin Packing in mehreren Dimensionen geben wir untere Schranken für das Kompetitivitätsverhältnis an und zeigen, dass die Ideen aus dem eindimensionalen Fall nicht direkt zu einer Verbesserung führen. Beim geometrischen Rucksackproblem ist es das Ziel, eine größtmögliche Teilmenge gegebener Rechtecke in einen einzelnen quadratischen Behälter zu packen. Für dieses Problem betrachten wir Approximationsalgorithmen. Für das Problem mit quadratischen Objekten verbessern wir die Laufzeit des bekannten PTAS zu einem EPTAS. Die langen Laufzeiten vieler Standardtechniken für geometrische Probleme können also vermieden werden. Schließlich zeigen wir, wie Ressourcenvergrößerung genutzt werden kann, um eine optimale Lösung in EPTAS-Zeit zu berechnen, was das bisherige PTAS verbessert.Google PhD Fellowshi
SLO-aware Colocation of Data Center Tasks Based on Instantaneous Processor Requirements
In a cloud data center, a single physical machine simultaneously executes
dozens of highly heterogeneous tasks. Such colocation results in more efficient
utilization of machines, but, when tasks' requirements exceed available
resources, some of the tasks might be throttled down or preempted. We analyze
version 2.1 of the Google cluster trace that shows short-term (1 second) task
CPU usage. Contrary to the assumptions taken by many theoretical studies, we
demonstrate that the empirical distributions do not follow any single
distribution. However, high percentiles of the total processor usage (summed
over at least 10 tasks) can be reasonably estimated by the Gaussian
distribution. We use this result for a probabilistic fit test, called the
Gaussian Percentile Approximation (GPA), for standard bin-packing algorithms.
To check whether a new task will fit into a machine, GPA checks whether the
resulting distribution's percentile corresponding to the requested service
level objective, SLO is still below the machine's capacity. In our simulation
experiments, GPA resulted in colocations exceeding the machines' capacity with
a frequency similar to the requested SLO.Comment: Author's version of a paper published in ACM SoCC'1
Packing Sporadic Real-Time Tasks on Identical Multiprocessor Systems
In real-time systems, in addition to the functional correctness recurrent
tasks must fulfill timing constraints to ensure the correct behavior of the
system. Partitioned scheduling is widely used in real-time systems, i.e., the
tasks are statically assigned onto processors while ensuring that all timing
constraints are met. The decision version of the problem, which is to check
whether the deadline constraints of tasks can be satisfied on a given number of
identical processors, has been known -complete in the strong sense.
Several studies on this problem are based on approximations involving resource
augmentation, i.e., speeding up individual processors. This paper studies
another type of resource augmentation by allocating additional processors, a
topic that has not been explored until recently. We provide polynomial-time
algorithms and analysis, in which the approximation factors are dependent upon
the input instances. Specifically, the factors are related to the maximum ratio
of the period to the relative deadline of a task in the given task set. We also
show that these algorithms unfortunately cannot achieve a constant
approximation factor for general cases. Furthermore, we prove that the problem
does not admit any asymptotic polynomial-time approximation scheme (APTAS)
unless when the task set has constrained deadlines, i.e.,
the relative deadline of a task is no more than the period of the task.Comment: Accepted and to appear in ISAAC 2018, Yi-Lan, Taiwa
Solving the Batch Stochastic Bin Packing Problem in Cloud: A Chance-constrained Optimization Approach
This paper investigates a critical resource allocation problem in the first
party cloud: scheduling containers to machines. There are tens of services and
each service runs a set of homogeneous containers with dynamic resource usage;
containers of a service are scheduled daily in a batch fashion. This problem
can be naturally formulated as Stochastic Bin Packing Problem (SBPP). However,
traditional SBPP research often focuses on cases of empty machines, whose
objective, i.e., to minimize the number of used machines, is not well-defined
for the more common reality with nonempty machines. This paper aims to close
this gap. First, we define a new objective metric, Used Capacity at Confidence
(UCaC), which measures the maximum used resources at a probability and is
proved to be consistent for both empty and nonempty machines, and reformulate
the SBPP under chance constraints. Second, by modeling the container resource
usage distribution in a generative approach, we reveal that UCaC can be
approximated with Gaussian, which is verified by trace data of real-world
applications. Third, we propose an exact solver by solving the equivalent
cutting stock variant as well as two heuristics-based solvers -- UCaC best fit,
bi-level heuristics. We experimentally evaluate these solvers on both synthetic
datasets and real application traces, demonstrating our methodology's advantage
over traditional SBPP optimal solver minimizing the number of used machines,
with a low rate of resource violations.Comment: To appear in SIGKDD 2022 as Research Track pape
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