98 research outputs found
Reconstruction Threshold for the Hardcore Model
In this paper we consider the reconstruction problem on the tree for the
hardcore model. We determine new bounds for the non-reconstruction regime on
the k-regular tree showing non-reconstruction when lambda < (ln
2-o(1))ln^2(k)/(2 lnln(k)) improving the previous best bound of lambda < e-1.
This is almost tight as reconstruction is known to hold when lambda>
(e+o(1))ln^2(k). We discuss the relationship for finding large independent sets
in sparse random graphs and to the mixing time of Markov chains for sampling
independent sets on trees.Comment: 14 pages, 2 figure
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
Reconstruction of Random Colourings
Reconstruction problems have been studied in a number of contexts including
biology, information theory and and statistical physics. We consider the
reconstruction problem for random -colourings on the -ary tree for
large . Bhatnagar et. al. showed non-reconstruction when and reconstruction when . We tighten this result and show non-reconstruction when and reconstruction when .Comment: Added references, updated notatio
Sampling Random Colorings of Sparse Random Graphs
We study the mixing properties of the single-site Markov chain known as the
Glauber dynamics for sampling -colorings of a sparse random graph
for constant . The best known rapid mixing results for general graphs are in
terms of the maximum degree of the input graph and hold when
for all . Improved results hold when for
graphs with girth and sufficiently large where is the root of ; further improvements on
the constant hold with stronger girth and maximum degree assumptions.
For sparse random graphs the maximum degree is a function of and the goal
is to obtain results in terms of the expected degree . The following rapid
mixing results for hold with high probability over the choice of the
random graph for sufficiently large constant~. Mossel and Sly (2009) proved
rapid mixing for constant , and Efthymiou (2014) improved this to linear
in~. The condition was improved to by Yin and Zhang (2016) using
non-MCMC methods. Here we prove rapid mixing when where
is the same constant as above. Moreover we obtain
mixing time of the Glauber dynamics, while in previous rapid mixing
results the exponent was an increasing function in . As in previous results
for random graphs our proof analyzes an appropriately defined block dynamics to
"hide" high-degree vertices. One new aspect in our improved approach is
utilizing so-called local uniformity properties for the analysis of block
dynamics. To analyze the "burn-in" phase we prove a concentration inequality
for the number of disagreements propagating in large blocks
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