2,925 research outputs found
Strong spatial mixing of list coloring of graphs
The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise known to be #P hard. In the context of coloring, strong spatial mixing has been established for Kelly trees in (Ge and Stefankovic, arXiv:1102.2886v3 (2011)) when q ≥ α[superscript *] Δ + 1 where q the number of colors, Δ is the degree and α[superscript *] is the unique solution to xe[superscript -1/x] = 1. It has also been established in (Goldberg et al., SICOMP 35 (2005) 486–517) for bounded degree lattice graphs whenever q ≥ α[superscript *] Δ - β for some constant β, where Δ is the maximum vertex degree of the graph. We establish strong spatial mixing for a more general problem, namely list coloring, for arbitrary bounded degree triangle-free graphs. Our results hold for any α > α[superscript *] whenever the size of the list of each vertex v is at least αΔ(v) + β where Δ(v) is the degree of vertex v and β is a constant that only depends on α. The result is obtained by proving the decay of correlations of marginal probabilities associated with graph nodes measured using a suitably chosen error function
Spatial Mixing of Coloring Random Graphs
We study the strong spatial mixing (decay of correlation) property of proper
-colorings of random graph with a fixed . The strong spatial
mixing of coloring and related models have been extensively studied on graphs
with bounded maximum degree. However, for typical classes of graphs with
bounded average degree, such as , an easy counterexample shows that
colorings do not exhibit strong spatial mixing with high probability.
Nevertheless, we show that for with and
sufficiently large , with high probability proper -colorings of
random graph exhibit strong spatial mixing with respect to an
arbitrarily fixed vertex. This is the first strong spatial mixing result for
colorings of graphs with unbounded maximum degree. Our analysis of strong
spatial mixing establishes a block-wise correlation decay instead of the
standard point-wise decay, which may be of interest by itself, especially for
graphs with unbounded degree
Rapid Mixing for Lattice Colorings with Fewer Colors
We provide an optimally mixing Markov chain for 6-colorings of the square
lattice on rectangular regions with free, fixed, or toroidal boundary
conditions. This implies that the uniform distribution on the set of such
colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet
has a finite correlation length and a unique Gibbs measure at zero temperature.
Four and five are now the only remaining values of q for which it is not known
whether there exists a rapidly mixing Markov chain for q-colorings of the
square lattice.Comment: Appeared in Proc. LATIN 2004, to appear in JSTA
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
- …