3,894 research outputs found
Online Multidimensional Packing Problems in the Random-Order Model
We study online multidimensional variants of the generalized assignment problem which are used to model prominent real-world applications, such as the assignment of virtual machines with multiple resource requirements to physical infrastructure in cloud computing. These problems can be seen as an extension of the well known secretary problem and thus the standard online worst-case model cannot provide any performance guarantee. The prevailing model in this case is the random-order model, which provides a useful realistic and robust alternative. Using this model, we study the d-dimensional generalized assignment problem, where we introduce a novel technique that achieves an O(d)-competitive algorithms and prove a matching lower bound of Omega(d). Furthermore, our algorithm improves upon the best-known competitive-ratio for the online (one-dimensional) generalized assignment problem and the online knapsack problem
Stochastic Combinatorial Optimization via Poisson Approximation
We study several stochastic combinatorial problems, including the expected
utility maximization problem, the stochastic knapsack problem and the
stochastic bin packing problem. A common technical challenge in these problems
is to optimize some function of the sum of a set of random variables. The
difficulty is mainly due to the fact that the probability distribution of the
sum is the convolution of a set of distributions, which is not an easy
objective function to work with. To tackle this difficulty, we introduce the
Poisson approximation technique. The technique is based on the Poisson
approximation theorem discovered by Le Cam, which enables us to approximate the
distribution of the sum of a set of random variables using a compound Poisson
distribution.
We first study the expected utility maximization problem introduced recently
[Li and Despande, FOCS11]. For monotone and Lipschitz utility functions, we
obtain an additive PTAS if there is a multidimensional PTAS for the
multi-objective version of the problem, strictly generalizing the previous
result.
For the stochastic bin packing problem (introduced in [Kleinberg, Rabani and
Tardos, STOC97]), we show there is a polynomial time algorithm which uses at
most the optimal number of bins, if we relax the size of each bin and the
overflow probability by eps.
For stochastic knapsack, we show a 1+eps-approximation using eps extra
capacity, even when the size and reward of each item may be correlated and
cancelations of items are allowed. This generalizes the previous work [Balghat,
Goel and Khanna, SODA11] for the case without correlation and cancelation. Our
algorithm is also simpler. We also present a factor 2+eps approximation
algorithm for stochastic knapsack with cancelations. the current known
approximation factor of 8 [Gupta, Krishnaswamy, Molinaro and Ravi, FOCS11].Comment: 42 pages, 1 figure, Preliminary version appears in the Proceeding of
the 45th ACM Symposium on the Theory of Computing (STOC13
Online Lower Bounds via Duality
In this paper, we exploit linear programming duality in the online setting
(i.e., where input arrives on the fly) from the unique perspective of designing
lower bounds on the competitive ratio. In particular, we provide a general
technique for obtaining online deterministic and randomized lower bounds (i.e.,
hardness results) on the competitive ratio for a wide variety of problems. We
show the usefulness of our approach by providing new, tight lower bounds for
three diverse online problems. The three problems we show tight lower bounds
for are the Vector Bin Packing problem, Ad-auctions (and various online
matching problems), and the Capital Investment problem. Our methods are
sufficiently general that they can also be used to reconstruct existing lower
bounds.
Our techniques are in stark contrast to previous works, which exploit linear
programming duality to obtain positive results, often via the useful
primal-dual scheme. We design a general recipe with the opposite aim of
obtaining negative results via duality. The general idea behind our approach is
to construct a primal linear program based on a collection of input sequences,
where the objective function corresponds to optimizing the competitive ratio.
We then obtain the corresponding dual linear program and provide a feasible
solution, where the objective function yields a lower bound on the competitive
ratio. Online lower bounds are often achieved by adapting the input sequence
according to an online algorithm's behavior and doing an appropriate ad hoc
case analysis. Using our unifying techniques, we simultaneously combine these
cases into one linear program and achieve online lower bounds via a more robust
analysis. We are confident that our framework can be successfully applied to
produce many more lower bounds for a wide array of online problems
Towards Bin Packing (preliminary problem survey, models with multiset estimates)
The paper described a generalized integrated glance to bin packing problems
including a brief literature survey and some new problem formulations for the
cases of multiset estimates of items. A new systemic viewpoint to bin packing
problems is suggested: (a) basic element sets (item set, bin set, item subset
assigned to bin), (b) binary relation over the sets: relation over item set as
compatibility, precedence, dominance; relation over items and bins (i.e.,
correspondence of items to bins). A special attention is targeted to the
following versions of bin packing problems: (a) problem with multiset estimates
of items, (b) problem with colored items (and some close problems). Applied
examples of bin packing problems are considered: (i) planning in paper industry
(framework of combinatorial problems), (ii) selection of information messages,
(iii) packing of messages/information packages in WiMAX communication system
(brief description).Comment: 39 pages, 18 figures, 14 table
Multi-resource Energy-efficient Routing in Cloud Data Centers with Networks-as-a-Service
With the rapid development of software defined networking and network
function virtualization, researchers have proposed a new cloud networking model
called Network-as-a-Service (NaaS) which enables both in-network packet
processing and application-specific network control. In this paper, we revisit
the problem of achieving network energy efficiency in data centers and identify
some new optimization challenges under the NaaS model. Particularly, we extend
the energy-efficient routing optimization from single-resource to
multi-resource settings. We characterize the problem through a detailed model
and provide a formal problem definition. Due to the high complexity of direct
solutions, we propose a greedy routing scheme to approximate the optimum, where
flows are selected progressively to exhaust residual capacities of active
nodes, and routing paths are assigned based on the distributions of both node
residual capacities and flow demands. By leveraging the structural regularity
of data center networks, we also provide a fast topology-aware heuristic method
based on hierarchically solving a series of vector bin packing instances. Our
simulations show that the proposed routing scheme can achieve significant gain
on energy savings and the topology-aware heuristic can produce comparably good
results while reducing the computation time to a large extent.Comment: 9 page
Robust Online Algorithms for Dynamic Problems
Online algorithms that allow a small amount of migration or recourse have
been intensively studied in the last years. They are essential in the design of
competitive algorithms for dynamic problems, where objects can also depart from
the instance. In this work, we give a general framework to obtain so called
robust online algorithms for these dynamic problems: these online algorithm
achieve an asymptotic competitive ratio of with migration
, where is the best known offline asymptotic
approximation ratio. In order to use our framework, one only needs to construct
a suitable online algorithm for the static online case, where items never
depart. To show the usefulness of our approach, we improve upon the best known
robust algorithms for the dynamic versions of generalizations of Strip Packing
and Bin Packing, including the first robust algorithm for general
-dimensional Bin Packing and Vector Packing
A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game
We prove a tight lower bound on the asymptotic performance ratio of
the bounded space online -hypercube bin packing problem, solving an open
question raised in 2005. In the classic -hypercube bin packing problem, we
are given a sequence of -dimensional hypercubes and we have an unlimited
number of bins, each of which is a -dimensional unit hypercube. The goal is
to pack (orthogonally) the given hypercubes into the minimum possible number of
bins, in such a way that no two hypercubes in the same bin overlap. The bounded
space online -hypercube bin packing problem is a variant of the
-hypercube bin packing problem, in which the hypercubes arrive online and
each one must be packed in an open bin without the knowledge of the next
hypercubes. Moreover, at each moment, only a constant number of open bins are
allowed (whenever a new bin is used, it is considered open, and it remains so
until it is considered closed, in which case, it is not allowed to accept new
hypercubes). Epstein and van Stee [SIAM J. Comput. 35 (2005), no. 2, 431-448]
showed that is and , and conjectured that
it is . We show that is in fact . To
obtain this result, we elaborate on some ideas presented by those authors, and
go one step further showing how to obtain better (offline) packings of certain
special instances for which one knows how many bins any bounded space algorithm
has to use. Our main contribution establishes the existence of such packings,
for large enough , using probabilistic arguments. Such packings also lead to
lower bounds for the prices of anarchy of the selfish -hypercube bin packing
game. We present a lower bound of for the pure price of
anarchy of this game, and we also give a lower bound of for
its strong price of anarchy
A service system with packing constraints: Greedy randomized algorithm achieving sublinear in scale optimality gap
A service system with multiple types of arriving customers is considered.
There is an infinite number of homogeneous servers. Multiple customers can be
placed for simultaneous service into one server, subject to general packing
constraints. Each new arriving customer is placed for service immediately,
either into an occupied server, as long as packing constraints are not
violated, or into an empty server. After service completion, each customer
leaves its server and the system. The basic objective is to minimize the number
of occupied servers in steady state. We study a Greedy-Random (GRAND) placement
(packing) algorithm, introduced in [23]. This is a simple online algorithm,
which places each arriving customer uniformly at random into either one of the
already occupied servers that can still fit the customer, or one of the
so-called zero-servers, which are empty servers designated to be available to
new arrivals. In [23], a version of the algorithm, labeled GRAND(), was
considered, where the number of zero servers is , with being the
current total number of customers in the system, and being an algorithm
parameter. GRAND() was shown in [23] to be asymptotically optimal in the
following sense: (a) the steady-state optimality gap grows linearly in the
system scale (the mean total number of customers in service), i.e. as for some ; and (b) as . In this paper, we
consider the GRAND() algorithm, in which the number of zero-servers is
, where is an algorithm parameter, and
is the maximum possible number of customers that a server can fit.
We prove the asymptotic optimality of GRAND() in the sense that the
steady-state optimality gap is , sublinear in the system scale. This is a
stronger form of asymptotic optimality than that of GRAND().Comment: Revision. 34 page
Competitive Algorithms from Competitive Equilibria: Non-Clairvoyant Scheduling under Polyhedral Constraints
We introduce and study a general scheduling problem that we term the Packing
Scheduling problem. In this problem, jobs can have different arrival times and
sizes; a scheduler can process job at rate , subject to arbitrary
packing constraints over the set of rates () of the outstanding jobs.
The PSP framework captures a variety of scheduling problems, including the
classical problems of unrelated machines scheduling, broadcast scheduling, and
scheduling jobs of different parallelizability. It also captures scheduling
constraints arising in diverse modern environments ranging from individual
computer architectures to data centers. More concretely, PSP models
multidimensional resource requirements and parallelizability, as well as
network bandwidth requirements found in data center scheduling.
In this paper, we design non-clairvoyant online algorithms for PSP and its
special cases -- in this setting, the scheduler is unaware of the sizes of
jobs. Our two main results are, 1) a constant competitive algorithm for
minimizing total weighted completion time for PSP and 2)a scalable algorithm
for minimizing the total flow-time on unrelated machines, which is a special
case of PSP.Comment: Accepted for publication in STOC 201
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