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 $(1 \pm \epsilon)$. Our first approach independently samples each edge $uv$ with probability inversely proportional to the edge-connectivity between $u$ and $v$. 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 $O(n \log^2(n)/\epsilon^2)$ 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