1,214 research outputs found
A New Upper Bound on 2D Online Bin Packing
The 2D Online Bin Packing is a fundamental problem in Computer Science and
the determination of its asymptotic competitive ratio has attracted great
research attention. In a long series of papers, the lower bound of this ratio
has been improved from 1.808, 1.856 to 1.907 and its upper bound reduced from
3.25, 3.0625, 2.8596, 2.7834 to 2.66013. In this paper, we rewrite the upper
bound record to 2.5545. Our idea for the improvement is as follows. In SODA
2002 \cite{SS03}, Seiden and van Stee proposed an elegant algorithm called , comprised of the {\em Harmonic algorithm} and the {\em Improved
Harmonic algorithm} , for the two-dimensional online bin packing problem and
proved that the algorithm has an asymptotic competitive ratio of at most
2.66013. Since the best known online algorithm for one-dimensional bin packing
is the {\em Super Harmonic algorithm} \cite{S02}, a natural question to ask is:
could a better upper bound be achieved by using the Super Harmonic algorithm
instead of the Improved Harmonic algorithm? However, as mentioned in
\cite{SS03}, the previous analysis framework does not work. In this paper, we
give a positive answer for the above question. A new upper bound of 2.5545 is
obtained for 2-dimensional online bin packing. The main idea is to develop new
weighting functions for the Super Harmonic algorithm and propose new techniques
to bound the total weight in a rectangular bin
Improved Lower Bounds for Online Hypercube and Rectangle Packing
Packing a given sequence of items into as few bins as possible in an online
fashion is a widely studied problem. We improve lower bounds for packing boxes
into bins in two or more dimensions, both for general algorithms for squares
and rectangles (in two dimensions) and for an important subclass, so-called
Harmonic-type algorithms for hypercubes (in two or more dimensions). Lastly, we
show that two adaptions of ideas from a one-dimensional packing algorithm to
square packing do not help to break the barrier of 2.Comment: 19 pages, 4 figure
Strip Packing vs. Bin Packing
In this paper we establish a general algorithmic framework between bin
packing and strip packing, with which we achieve the same asymptotic bounds by
applying bin packing algorithms to strip packing. More precisely we obtain the
following results: (1) Any offline bin packing algorithm can be applied to
strip packing maintaining the same asymptotic worst-case ratio. Thus using FFD
(MFFD) as a subroutine, we get a practical (simple and fast) algorithm for
strip packing with an upper bound 11/9 (71/60). A simple AFPTAS for strip
packing immediately follows. (2) A class of Harmonic-based algorithms for bin
packing can be applied to online strip packing maintaining the same asymptotic
competitive ratio. It implies online strip packing admits an upper bound of
1.58889 on the asymptotic competitive ratio, which is very close to the lower
bound 1.5401 and significantly improves the previously best bound of 1.6910 and
affirmatively answers an open question posed by Csirik et. al.Comment: 12 pages, 3 figure
Improved online hypercube packing
In this paper, we study online multidimensional bin packing problem when all
items are hypercubes.
Based on the techniques in one dimensional bin packing algorithm Super
Harmonic by Seiden, we give a framework for online hypercube packing problem
and obtain new upper bounds of asymptotic competitive ratios.
For square packing, we get an upper bound of 2.1439, which is better than
2.24437.
For cube packing, we also give a new upper bound 2.6852 which is better than
2.9421 by Epstein and van Stee.Comment: 13 pages, one figure, accepted in WAOA'0
New Upper Bounds on The Approximability of 3D Strip Packing
In this paper, we study the 3D strip packing problem in which we are given a
list of 3-dimensional boxes and required to pack all of them into a
3-dimensional strip with length 1 and width 1 and unlimited height to minimize
the height used. Our results are below: i) we give an approximation algorithm
with asymptotic worst-case ratio 1.69103, which improves the previous best
bound of by Jansen and Solis-Oba of SODA 2006; ii) we also present
an asymptotic PTAS for the case in which all items have {\em square} bases.Comment: Submitted to SODA 200
Fully-Dynamic Bin Packing with Limited Repacking
We study the classic Bin Packing problem in a fully-dynamic setting, where
new items can arrive and old items may depart. We want algorithms with low
asymptotic competitive ratio \emph{while repacking items sparingly} between
updates. Formally, each item has a \emph{movement cost} , and we
want to use bins and incur a movement cost , either in the worst case, or in an amortized sense, for
as small as possible. We call the \emph{recourse} of the algorithm.
This is motivated by cloud storage applications, where fully-dynamic Bin
Packing models the problem of data backup to minimize the number of disks used,
as well as communication incurred in moving file backups between disks. Since
the set of files changes over time, we could recompute a solution periodically
from scratch, but this would give a high number of disk rewrites, incurring a
high energy cost and possible wear and tear of the disks. In this work, we
present optimal tradeoffs between number of bins used and number of items
repacked, as well as natural extensions of the latter measure.Comment: To appear in ICALP 2018. Improved worst-case recourse for unit costs
added (Theorem 2.7
Beating the Harmonic lower bound for online bin packing
In the online bin packing problem, items of sizes in (0,1] arrive online to
be packed into bins of size 1. The goal is to minimize the number of used bins.
In this paper, we present an online bin packing algorithm with asymptotic
competitive ratio of 1.5813. This is the first improvement in fifteen years and
reduces the gap to the lower bound by 15%. Within the well-known SuperHarmonic
framework, no competitive ratio below 1.58333 can be achieved.
We make two crucial changes to that framework. First, some of our algorithm's
decisions depend on exact sizes of items, instead of only their types. In
particular, for each item with size in (1/3,1/2], we use its exact size to
determine if it can be packed together with an item of size greater than 1/2.
Second, we add constraints to the linear programs considered by Seiden, in
order to better lower bound the optimal solution. These extra constraints are
based on marks that we give to items based on how they are packed by our
algorithm. We show that for each input, a single weighting function can be
constructed to upper bound the competitive ratio on it.
We use this idea to simplify the analysis of SuperHarmonic, and show that the
algorithm Harmonic++ is in fact 1.58880-competitive (Seiden proved 1.58889),
and that 1.5884 can be achieved within the SuperHarmonic framework. Finally, we
give a lower bound of 1.5762 for our new framework.Comment: Added reference and clarified some sentence
Efficient Online Strategies for Renting Servers in the Cloud
In Cloud systems, we often deal with jobs that arrive and depart in an online
manner. Upon its arrival, a job should be assigned to a server. Each job has a
size which defines the amount of resources that it needs. Servers have uniform
capacity and, at all times, the total size of jobs assigned to a server should
not exceed the capacity. This setting is closely related to the classic bin
packing problem. The difference is that, in bin packing, the objective is to
minimize the total number of used servers. In the Cloud, however, the charge
for each server is proportional to the length of the time interval it is rented
for, and the goal is to minimize the cost involved in renting all used servers.
Recently, certain bin packing strategies were considered for renting servers in
the Cloud [Li et al. SPAA'14]. There, it is proved that all Any-Fit bin packing
strategy has a competitive ratio of at least , where is the max/min
interval length ratio of jobs. It is also shown that First Fit has a
competitive ratio of while Best Fit is not competitive at all. We
observe that the lower bound of extends to all online algorithms. We also
prove that, surprisingly, Next Fit algorithm has competitive ratio of at most
. We also show that a variant of Next Fit achieves a competitive
ratio of , where is a parameter of the
algorithm. In particular, if the value of is known, the algorithm has a
competitive ratio of ; this improves upon the existing upper bound of
. Finally, we introduce a simple algorithm called Move To Front (MTF)
which has a competitive ratio of at most and also promising
average-case performance. We experimentally study the average-case performance
of different algorithms and observe that the typical behaviour of MTF is
distinctively better than other algorithms.Comment: 13 pages, 3 figure
Online bin packing with cardinality constraints revisited
Bin packing with cardinality constraints is a bin packing problem where an
upper bound k \geq 2 on the number of items packed into each bin is given, in
addition to the standard constraint on the total size of items packed into a
bin. We study the online scenario where items are presented one by one. We
analyze it with respect to the absolute competitive ratio and prove tight
bounds of 2 for any k \geq 4. We show that First Fit also has an absolute
competitive ratio of 2 for k=4, but not for larger values of k, and we present
a complete analysis of its asymptotic competitive ratio for all values of k
\geq 5. Additionally, we study the case of small with respect to the
asymptotic competitive ratio and the absolute competitive ratio
A new and improved algorithm for online bin packing
We revisit the classic online bin packing problem. In this problem, items of
positive sizes no larger than 1 are presented one by one to be packed into
subsets called "bins" of total sizes no larger than 1, such that every item is
assigned to a bin before the next item is presented. We use online partitioning
of items into classes based on sizes, as in previous work, but we also apply a
new method where items of one class can be packed into more than two types of
bins, where a bin type is defined according to the number of such items grouped
together. Additionally, we allow the smallest class of items to be packed in
multiple kinds of bins, and not only into their own bins. We combine this with
the approach of packing of sufficiently big items according to their exact
sizes. Finally, we simplify the analysis of such algorithms, allowing the
analysis to be based on the most standard weight functions. This simplified
analysis allows us to study the algorithm which we defined based on all these
ideas. This leads us to the design and analysis of the first algorithm of
asymptotic competitive ratio strictly below 1.58, specifically, we break this
barrier and provide an algorithm AH (Advanced Harmonic) whose asymptotic
competitive ratio does not exceed 1.5783
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