276 research outputs found
Rainbow perfect matchings in r-partite graph structures
A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.Peer ReviewedPostprint (author's final draft
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
Shapes of interacting RNA complexes
Shapes of interacting RNA complexes are studied using a filtration via their
topological genus. A shape of an RNA complex is obtained by (iteratively)
collapsing stacks and eliminating hairpin loops. This shape-projection
preserves the topological core of the RNA complex and for fixed topological
genus there are only finitely many such shapes.Our main result is a new
bijection that relates the shapes of RNA complexes with shapes of RNA
structures.This allows to compute the shape polynomial of RNA complexes via the
shape polynomial of RNA structures. We furthermore present a linear time
uniform sampling algorithm for shapes of RNA complexes of fixed topological
genus.Comment: 38 pages 24 figure
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