2,982 research outputs found
Hybrid next-fit algorithm for the two-dimensional rectangle bin-packing problem
We present a new approximation algorithm for the two-dimensional bin-packing problem. The algorithm is based on two one-dimensional bin-packing algorithms. Since the algorithm is of next-fit type it can also be used for those cases where the output is required to be on-line (e. g. if we open an new bin we have no possibility to pack elements into the earlier opened bins). We give a tight bound for its worst-case and show that this bound is a parameter of the maximal sizes of the items to be packed. Moreover, we also present a probabilistic analysis of this algorithm.worst-case analysis;probabilistic analysis;bin-packing;heuristic algorithm;on-line algorithm;two-dimensional packing
Overcommitment in Cloud Services -- Bin packing with Chance Constraints
This paper considers a traditional problem of resource allocation, scheduling
jobs on machines. One such recent application is cloud computing, where jobs
arrive in an online fashion with capacity requirements and need to be
immediately scheduled on physical machines in data centers. It is often
observed that the requested capacities are not fully utilized, hence offering
an opportunity to employ an overcommitment policy, i.e., selling resources
beyond capacity. Setting the right overcommitment level can induce a
significant cost reduction for the cloud provider, while only inducing a very
low risk of violating capacity constraints. We introduce and study a model that
quantifies the value of overcommitment by modeling the problem as a bin packing
with chance constraints. We then propose an alternative formulation that
transforms each chance constraint into a submodular function. We show that our
model captures the risk pooling effect and can guide scheduling and
overcommitment decisions. We also develop a family of online algorithms that
are intuitive, easy to implement and provide a constant factor guarantee from
optimal. Finally, we calibrate our model using realistic workload data, and
test our approach in a practical setting. Our analysis and experiments
illustrate the benefit of overcommitment in cloud services, and suggest a cost
reduction of 1.5% to 17% depending on the provider's risk tolerance
Two-dimensional rectangle packing: on-line methods and results
The first algorithms for the on-line two-dimensional rectangle packing problem were introduced by Coppersmith and Raghavan. They showed that for a family of heuristics 13/4 is an upper bound for the asymptotic worst-case ratios. We have investigated the Next Fit and the First Fit variants of their method. We proved that the asymptotic worst-case ratio equals 13/4 for the Next Fit variant and that 49/16 is an upper bound of the asymptotic worst-case ratio for the First Fit variant.
Online Computation with Untrusted Advice
The advice model of online computation captures a setting in which the
algorithm is given some partial information concerning the request sequence.
This paradigm allows to establish tradeoffs between the amount of this
additional information and the performance of the online algorithm. However, if
the advice is corrupt or, worse, if it comes from a malicious source, the
algorithm may perform poorly. In this work, we study online computation in a
setting in which the advice is provided by an untrusted source. Our objective
is to quantify the impact of untrusted advice so as to design and analyze
online algorithms that are robust and perform well even when the advice is
generated in a malicious, adversarial manner. To this end, we focus on
well-studied online problems such as ski rental, online bidding, bin packing,
and list update. For ski-rental and online bidding, we show how to obtain
algorithms that are Pareto-optimal with respect to the competitive ratios
achieved; this improves upon the framework of Purohit et al. [NeurIPS 2018] in
which Pareto-optimality is not necessarily guaranteed. For bin packing and list
update, we give online algorithms with worst-case tradeoffs in their
competitiveness, depending on whether the advice is trusted or not; this is
motivated by work of Lykouris and Vassilvitskii [ICML 2018] on the paging
problem, but in which the competitiveness depends on the reliability of the
advice. Furthermore, we demonstrate how to prove lower bounds, within this
model, on the tradeoff between the number of advice bits and the
competitiveness of any online algorithm. Last, we study the effect of
randomization: here we show that for ski-rental there is a randomized algorithm
that Pareto-dominates any deterministic algorithm with advice of any size. We
also show that a single random bit is not always inferior to a single advice
bit, as it happens in the standard model
Locality-preserving allocations Problems and coloured Bin Packing
We study the following problem, introduced by Chung et al. in 2006. We are
given, online or offline, a set of coloured items of different sizes, and wish
to pack them into bins of equal size so that we use few bins in total (at most
times optimal), and that the items of each colour span few bins (at
most times optimal). We call such allocations -approximate. As usual in bin packing problems, we allow additive
constants and consider as the asymptotic performance ratios.
We prove that for \eps>0, if we desire small , no scheme can beat
(1+\eps, \Omega(1/\eps))-approximate allocations and similarly as we desire
small , no scheme can beat (1.69103, 1+\eps)-approximate allocations.
We give offline schemes that come very close to achieving these lower bounds.
For the online case, we prove that no scheme can even achieve
-approximate allocations. However, a small restriction on item
sizes permits a simple online scheme that computes (2+\eps, 1.7)-approximate
allocations
Generalized selfish bin packing
Standard bin packing is the problem of partitioning a set of items with
positive sizes no larger than 1 into a minimum number of subsets (called bins)
each having a total size of at most 1. In bin packing games, an item has a
positive weight, and given a valid packing or partition of the items, each item
has a cost or a payoff associated with it. We study a class of bin packing
games where the payoff of an item is the ratio between its weight and the total
weight of items packed with it, that is, the cost sharing is based linearly on
the weights of items. We study several types of pure Nash equilibria: standard
Nash equilibria, strong equilibria, strictly Pareto optimal equilibria, and
weakly Pareto optimal equilibria. We show that any game of this class admits
all these types of equilibria. We study the (asymptotic) prices of anarchy and
stability (PoA and PoS) of the problem with respect to these four types of
equilibria, for the two cases of general weights and of unit weights. We show
that while the case of general weights is strongly related to the well-known
First Fit algorithm, and all the four PoA values are equal to 1.7, this is not
true for unit weights. In particular, we show that all of them are strictly
below 1.7, the strong PoA is equal to approximately 1.691 (another well-known
number in bin packing) while the strictly Pareto optimal PoA is much lower. We
show that all the PoS values are equal to 1, except for those of strong
equilibria, which is equal to 1.7 for general weights, and to approximately
1.611824 for unit weights. This last value is not known to be the (asymptotic)
approximation ratio of any well-known algorithm for bin packing. Finally, we
study convergence to equilibria
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