3,182 research outputs found
Approximating the MaxCover Problem with Bounded Frequencies in FPT Time
We study approximation algorithms for several variants of the MaxCover
problem, with the focus on algorithms that run in FPT time. In the MaxCover
problem we are given a set N of elements, a family S of subsets of N, and an
integer K. The goal is to find up to K sets from S that jointly cover (i.e.,
include) as many elements as possible. This problem is well-known to be NP-hard
and, under standard complexity-theoretic assumptions, the best possible
polynomial-time approximation algorithm has approximation ratio (1 - 1/e). We
first consider a variant of MaxCover with bounded element frequencies, i.e., a
variant where there is a constant p such that each element belongs to at most p
sets in S. For this case we show that there is an FPT approximation scheme
(i.e., for each B there is a B-approximation algorithm running in FPT time) for
the problem of maximizing the number of covered elements, and a randomized FPT
approximation scheme for the problem of minimizing the number of elements left
uncovered (we take K to be the parameter). Then, for the case where there is a
constant p such that each element belongs to at least p sets from S, we show
that the standard greedy approximation algorithm achieves approximation ratio
exactly (1-e^{-max(pK/|S|, 1)}). We conclude by considering an unrestricted
variant of MaxCover, and show approximation algorithms that run in exponential
time and combine an exact algorithm with a greedy approximation. Some of our
results improve currently known results for MaxVertexCover
Exploring heterogeneity of unreliable machines for p2p backup
P2P architecture is a viable option for enterprise backup. In contrast to
dedicated backup servers, nowadays a standard solution, making backups directly
on organization's workstations should be cheaper (as existing hardware is
used), more efficient (as there is no single bottleneck server) and more
reliable (as the machines are geographically dispersed).
We present the architecture of a p2p backup system that uses pairwise
replication contracts between a data owner and a replicator. In contrast to
standard p2p storage systems using directly a DHT, the contracts allow our
system to optimize replicas' placement depending on a specific optimization
strategy, and so to take advantage of the heterogeneity of the machines and the
network. Such optimization is particularly appealing in the context of backup:
replicas can be geographically dispersed, the load sent over the network can be
minimized, or the optimization goal can be to minimize the backup/restore time.
However, managing the contracts, keeping them consistent and adjusting them in
response to dynamically changing environment is challenging.
We built a scientific prototype and ran the experiments on 150 workstations
in the university's computer laboratories and, separately, on 50 PlanetLab
nodes. We found out that the main factor affecting the quality of the system is
the availability of the machines. Yet, our main conclusion is that it is
possible to build an efficient and reliable backup system on highly unreliable
machines (our computers had just 13% average availability)
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