869,234 research outputs found
Optimal Distributed Covering Algorithms
We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is (f+epsilon). Let Delta denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log{Delta} / log log Delta) rounds, for constants epsilon in (0,1] and f in N^+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms.
For constant values of f and epsilon, our algorithm improves over the (f+epsilon)-approximation algorithm of [Fabian Kuhn et al., 2006] whose running time is O(log Delta + log W), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in O(f log n) rounds, improving over the classical result of [Samir Khuller et al., 1994] that achieves a running time of O(f log^2 n). Finally, for weighted vertex cover (f=2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of [Koufogiannakis and Young, 2011].
We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (f+epsilon)-approximate integral solution in O((1+f/log n)* ((log Delta)/(log log Delta) + (f * log M)^{1.01}* log epsilon^{-1}* (log Delta)^{0.01})) rounds, where f bounds the number of variables in a constraint, Delta bounds the number of constraints a variable appears in, and M=max {1, ceil[1/a_{min}]}, where a_{min} is the smallest normalized constraint coefficient. This improves over the results of [Fabian Kuhn et al., 2006] for the integral case, which combined with rounding achieves the same guarantees in O(epsilon^{-4}* f^4 * log f * log(M * Delta)) rounds
Optimal Recombination in Genetic Algorithms
This paper surveys results on complexity of the optimal recombination problem
(ORP), which consists in finding the best possible offspring as a result of a
recombination operator in a genetic algorithm, given two parent solutions. We
consider efficient reductions of the ORPs, allowing to establish polynomial
solvability or NP-hardness of the ORPs, as well as direct proofs of hardness
results
Greedy Algorithms for Optimal Distribution Approximation
The approximation of a discrete probability distribution by an
-type distribution is considered. The approximation error is
measured by the informational divergence
, which is an appropriate measure, e.g.,
in the context of data compression. Properties of the optimal approximation are
derived and bounds on the approximation error are presented, which are
asymptotically tight. It is shown that -type approximations that minimize
either , or
, or the variational distance
can all be found by using specific
instances of the same general greedy algorithm.Comment: 5 page
Optimal Placement Algorithms for Virtual Machines
Cloud computing provides a computing platform for the users to meet their
demands in an efficient, cost-effective way. Virtualization technologies are
used in the clouds to aid the efficient usage of hardware. Virtual machines
(VMs) are utilized to satisfy the user needs and are placed on physical
machines (PMs) of the cloud for effective usage of hardware resources and
electricity in the cloud. Optimizing the number of PMs used helps in cutting
down the power consumption by a substantial amount.
In this paper, we present an optimal technique to map virtual machines to
physical machines (nodes) such that the number of required nodes is minimized.
We provide two approaches based on linear programming and quadratic programming
techniques that significantly improve over the existing theoretical bounds and
efficiently solve the problem of virtual machine (VM) placement in data
centers
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