19,154 research outputs found
Robust randomized matchings
The following game is played on a weighted graph: Alice selects a matching
and Bob selects a number . Alice's payoff is the ratio of the weight of
the heaviest edges of to the maximum weight of a matching of size at
most . If guarantees a payoff of at least then it is called
-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns
a -robust matching, which is best possible.
We show that Alice can improve her payoff to by playing a
randomized strategy. This result extends to a very general class of
independence systems that includes matroid intersection, b-matchings, and
strong 2-exchange systems. It also implies an improved approximation factor for
a stochastic optimization variant known as the maximum priority matching
problem and translates to an asymptotic robustness guarantee for deterministic
matchings, in which Bob can only select numbers larger than a given constant.
Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound
Community detection and stochastic block models: recent developments
The stochastic block model (SBM) is a random graph model with planted
clusters. It is widely employed as a canonical model to study clustering and
community detection, and provides generally a fertile ground to study the
statistical and computational tradeoffs that arise in network and data
sciences.
This note surveys the recent developments that establish the fundamental
limits for community detection in the SBM, both with respect to
information-theoretic and computational thresholds, and for various recovery
requirements such as exact, partial and weak recovery (a.k.a., detection). The
main results discussed are the phase transitions for exact recovery at the
Chernoff-Hellinger threshold, the phase transition for weak recovery at the
Kesten-Stigum threshold, the optimal distortion-SNR tradeoff for partial
recovery, the learning of the SBM parameters and the gap between
information-theoretic and computational thresholds.
The note also covers some of the algorithms developed in the quest of
achieving the limits, in particular two-round algorithms via graph-splitting,
semi-definite programming, linearized belief propagation, classical and
nonbacktracking spectral methods. A few open problems are also discussed
More about vanishing cycles and mutation
The paper continues the discussion of symplectic aspects of Picard-Lefschetz
theory begun in "Vanishing cycles and mutation" (this archive). There we
explained how to associate to a suitable fibration over a two-dimensional disc
a triangulated category, the "derived directed Fukaya category" which describes
the structure of the vanishing cycles. The present second part serves two
purposes. Firstly, it contains various kinds of algebro-geometric examples,
including the "mirror manifold" of the projective plane. Secondly there is a
(largely conjectural) discussion of more advanced topics, such as (i)
Hochschild cohomology, (ii) relations between Picard-Lefschetz theory and Morse
theory, (iii) a proposed "dimensional reduction" algorithm for doing certain
Floer cohomology computations.Comment: 33 pages, LaTeX2e, 9 eps figure
An asymptotically optimal push-pull method for multicasting over a random network
We consider allcast and multicast flow problems where either all of the nodes
or only a subset of the nodes may be in session. Traffic from each node in the
session has to be sent to every other node in the session. If the session does
not consist of all the nodes, the remaining nodes act as relays. The nodes are
connected by undirected links whose capacities are independent and identically
distributed random variables. We study the asymptotics of the capacity region
(with network coding) in the limit of a large number of nodes, and show that
the normalized sum rate converges to a constant almost surely. We then provide
a decentralized push-pull algorithm that asymptotically achieves this
normalized sum rate without network coding.Comment: 13 pages, extended version of paper presented at the IEEE
International Symposium on Information Theory (ISIT) 2012, minor revision to
text to address review comments, to appear in IEEE Transactions in
information theor
A transfer matrix approach to the enumeration of plane meanders
A closed plane meander of order is a closed self-avoiding curve
intersecting an infinite line times. Meanders are considered distinct up
to any smooth deformation leaving the line fixed. We have developed an improved
algorithm, based on transfer matrix methods, for the enumeration of plane
meanders. While the algorithm has exponential complexity, its rate of growth is
much smaller than that of previous algorithms. The algorithm is easily modified
to enumerate various systems of closed meanders, semi-meanders, open meanders
and many other geometries.Comment: 13 pages, 9 eps figures, to appear in J. Phys.
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