929 research outputs found
Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach
In this paper, we study the -forest problem in the model of resource
augmentation. In the -forest problem, given an edge-weighted graph ,
a parameter , and a set of demand pairs , the
objective is to construct a minimum-cost subgraph that connects at least
demands. The problem is hard to approximate---the best-known approximation
ratio is . Furthermore, -forest is as hard to
approximate as the notoriously-hard densest -subgraph problem.
While the -forest problem is hard to approximate in the worst-case, we
show that with the use of resource augmentation, we can efficiently approximate
it up to a constant factor.
First, we restate the problem in terms of the number of demands that are {\em
not} connected. In particular, the objective of the -forest problem can be
viewed as to remove at most demands and find a minimum-cost subgraph that
connects the remaining demands. We use this perspective of the problem to
explain the performance of our algorithm (in terms of the augmentation) in a
more intuitive way.
Specifically, we present a polynomial-time algorithm for the -forest
problem that, for every , removes at most demands and has
cost no more than times the cost of an optimal algorithm
that removes at most demands
Lagrangian Duality based Algorithms in Online Energy-Efficient Scheduling
We study online scheduling problems in the general energy model of speed scaling with power down. The latter is a combination of the two extensively studied energy models, speed scaling and power down, toward a more realistic one. Due to the limits of the current techniques, only few results have been known in the general energy model in contrast to the large literature of the previous ones.
In the paper, we consider a Lagrangian duality based approach to design and analyze algorithms in the general energy model. We show the applicability of the approach to problems which are unlikely to admit a convex relaxation. Specifically, we consider the problem of minimizing energy with a single machine in which jobs arrive online and have to be processed before their deadlines. We present an alpha^alpha-competitive algorithm (whose the analysis is tight up to a constant factor) where the energy power function is of typical form z^alpha + g for constants alpha > 2 and g non-negative. Besides, we also consider the problem of minimizing the weighted flow-time plus energy. We give an O(alpha/ln(alpha))-competitive algorithm; that matches (up to a constant factor) to the currently best known algorithm for this problem in the restricted model of speed scaling
Online Primal-Dual For Non-linear Optimization with Applications to Speed Scaling
We reinterpret some online greedy algorithms for a class of nonlinear
"load-balancing" problems as solving a mathematical program online. For
example, we consider the problem of assigning jobs to (unrelated) machines to
minimize the sum of the alpha^{th}-powers of the loads plus assignment costs
(the online Generalized Assignment Problem); or choosing paths to connect
terminal pairs to minimize the alpha^{th}-powers of the edge loads (online
routing with speed-scalable routers). We give analyses of these online
algorithms using the dual of the primal program as a lower bound for the
optimal algorithm, much in the spirit of online primal-dual results for linear
problems.
We then observe that a wide class of uni-processor speed scaling problems
(with essentially arbitrary scheduling objectives) can be viewed as such load
balancing problems with linear assignment costs. This connection gives new
algorithms for problems that had resisted solutions using the dominant
potential function approaches used in the speed scaling literature, as well as
alternate, cleaner proofs for other known results
Lagrangian Duality in Online Scheduling with Resource Augmentation and Speed Scaling
International audienceWe present an unified approach to study online scheduling problems in the resource augmentation/speed scaling models. Potential function method is extensively used for analyzing algorithms in these models; however, they yields little insight on how to construct potential functions and how to design algorithms for related problems. In the paper, we generalize and strengthen the dual-fitting technique proposed by Anand et al. [1]. The approach consists of considering a possibly non-convex relaxation and its Lagrangian dual; then constructing dual variables such that the Lagrangian dual has objective value within a desired factor of the primal optimum. The competitive ratio follows by the standard Lagrangian weak duality. This approach is simple yet powerful and it is seemingly a right tool to study problems with resource augmentation or speed scaling. We illustrate the approach through the following results
Online Energy Storage Management: an Algorithmic Approach
Motivated by the importance of energy storage networks in smart grids, we provide an algorithmic study of the online energy storage management problem in a network setting, the first to the best of our knowledge. Given online power supplies, either entirely renewable supplies or those in combination with traditional supplies, we want to route power from the supplies to demands using storage units subject to a decay factor. Our goal is to maximize the total utility of satisfied demands less the total production cost of routed power. We model renewable supplies with the zero production cost function and traditional supplies with convex production cost functions. For two natural storage unit settings, private and public, we design poly-logarithmic competitive algorithms in the network flow model using the dual fitting and online primal dual methods for convex problems. Furthermore, we show strong hardness results for more general settings of the problem. Our techniques may be of independent interest in other routing and storage management problems
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