4,309 research outputs found
An Algorithmic Proof of the Lovasz Local Lemma via Resampling Oracles
The Lovasz Local Lemma is a seminal result in probabilistic combinatorics. It
gives a sufficient condition on a probability space and a collection of events
for the existence of an outcome that simultaneously avoids all of those events.
Finding such an outcome by an efficient algorithm has been an active research
topic for decades. Breakthrough work of Moser and Tardos (2009) presented an
efficient algorithm for a general setting primarily characterized by a product
structure on the probability space.
In this work we present an efficient algorithm for a much more general
setting. Our main assumption is that there exist certain functions, called
resampling oracles, that can be invoked to address the undesired occurrence of
the events. We show that, in all scenarios to which the original Lovasz Local
Lemma applies, there exist resampling oracles, although they are not
necessarily efficient. Nevertheless, for essentially all known applications of
the Lovasz Local Lemma and its generalizations, we have designed efficient
resampling oracles. As applications of these techniques, we present new results
for packings of Latin transversals, rainbow matchings and rainbow spanning
trees.Comment: 47 page
Graph Sparsification by Edge-Connectivity and Random Spanning Trees
We present new approaches to constructing graph sparsifiers --- weighted
subgraphs for which every cut has the same value as the original graph, up to a
factor of . Our first approach independently samples each
edge with probability inversely proportional to the edge-connectivity
between and . The fact that this approach produces a sparsifier resolves
a question posed by Bencz\'ur and Karger (2002). Concurrent work of Hariharan
and Panigrahi also resolves this question. Our second approach constructs a
sparsifier by forming the union of several uniformly random spanning trees.
Both of our approaches produce sparsifiers with
edges. Our proofs are based on extensions of Karger's contraction algorithm,
which may be of independent interest
Science with the Constellation-X Observatory
The Constellation X-ray Mission is a high throughput X-ray facility
emphasizing observations at high spectral resolution (E/\Delta E \sim
300-3000), and broad energy bandpass (0.25-40 keV). Constellation-X will
provide a factor of nearly 100 increase in sensitivity over current high
resolution X-ray spectroscopy missions. It is the X-ray astronomy equivalent of
large ground-based optical telescopes such as the Keck Observatory and the ESO
Very Large Telescope. When observations commence toward the end of next decade,
Constellation-X will address many fundamental astrophysics questions such as:
the formation and evolution of clusters of galaxies; constraining the baryon
content of the Universe; determining the spin and mass of supermassive black
holes in AGN; and probing strong gravity in the vicinity of black holes.Comment: to appear in "After the Dark Ages: When Galaxies Were Young", eds.
S.S. Holt and E.P. Smith, 4 pages, 1 figur
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