77 research outputs found
A sufficiently fast algorithm for finding close to optimal clique trees
AbstractWe offer an algorithm that finds a clique tree such that the size of the largest clique is at most (2α+1)k where k is the size of the largest clique in a clique tree in which this size is minimized and α is the approximation ratio of an α-approximation algorithm for the 3-way vertex cut problem. When α=4/3, our algorithm's complexity is O(24.67kn·poly(n)) and it errs by a factor of 3.67 where poly(n) is the running time of linear programming. This algorithm is extended to find clique trees in which the state space of the largest clique is bounded. When k=O(logn), our algorithm yields a polynomial inference algorithm for Bayesian networks
High-Dimensional Lipschitz Functions are Typically Flat
A homomorphism height function on the -dimensional torus
is a function taking integer values on the vertices of the torus with
consecutive integers assigned to adjacent vertices. A Lipschitz height function
is defined similarly but may also take equal values on adjacent vertices. In
each model, we consider the uniform distribution over such functions, subject
to boundary conditions. We prove that in high dimensions, with zero boundary
values, a typical function is very flat, having bounded variance at any fixed
vertex and taking at most values with high probability. Our
results extend to any dimension , if is replaced by an
enhanced version of it, the torus for
some fixed . This establishes one side of a conjectured roughening
transition in dimensions. The full transition is established for a class of
tori with non-equal side lengths. We also find that when is taken to
infinity while remains fixed, a typical function takes at most values
with high probability, where for the homomorphism model and for the
Lipschitz model. Suitable generalizations are obtained when grows with .
Our results apply also to the related model of uniform 3-coloring and
establish, for certain boundary conditions, that a uniformly sampled proper
3-coloring of will be nearly constant on either the even or
odd sub-lattice.
Our proofs are based on a combinatorial transformation and on a careful
analysis of the properties of a class of cutsets which we term odd cutsets. For
the Lipschitz model, our results rely also on a bijection of Yadin. This work
generalizes results of Galvin and Kahn, refutes a conjecture of Benjamini,
Yadin and Yehudayoff and answers a question of Benjamini, H\"aggstr\"om and
Mossel.Comment: 63 pages, 5 figures (containing 10 images). Improved introduction and
layout. Minor correction
Entropy-Driven Phase Transition in Low-Temperature Antiferromagnetic Potts Models
We prove the existence of long-range order at sufficiently low temperatures, including zero temperature, for the three-state Potts antiferromagnet on a class of quasi-transitive plane quadrangulations, including the diced lattice. More precisely, we show the existence of (at least) three infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one sublattice have a higher probability to be in one state than in either of the other two states. For the special case of the diced lattice, we give a good rigorous lower bound on this probability, based on computer-assisted calculations that are not available for the other lattices
Lecture Notes on Network Information Theory
These lecture notes have been converted to a book titled Network Information
Theory published recently by Cambridge University Press. This book provides a
significantly expanded exposition of the material in the lecture notes as well
as problems and bibliographic notes at the end of each chapter. The authors are
currently preparing a set of slides based on the book that will be posted in
the second half of 2012. More information about the book can be found at
http://www.cambridge.org/9781107008731/. The previous (and obsolete) version of
the lecture notes can be found at http://arxiv.org/abs/1001.3404v4/
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