212 research outputs found
Fractional total colourings of graphs of high girth
Reed conjectured that for every epsilon>0 and Delta there exists g such that
the fractional total chromatic number of a graph with maximum degree Delta and
girth at least g is at most Delta+1+epsilon. We prove the conjecture for
Delta=3 and for even Delta>=4 in the following stronger form: For each of these
values of Delta, there exists g such that the fractional total chromatic number
of any graph with maximum degree Delta and girth at least g is equal to
Delta+1
Metric Construction, Stopping Times and Path Coupling
In this paper we examine the importance of the choice of metric in path
coupling, and the relationship of this to \emph{stopping time analysis}. We
give strong evidence that stopping time analysis is no more powerful than
standard path coupling. In particular, we prove a stronger theorem for path
coupling with stopping times, using a metric which allows us to restrict
analysis to standard one-step path coupling. This approach provides insight for
the design of non-standard metrics giving improvements in the analysis of
specific problems.
We give illustrative applications to hypergraph independent sets and SAT
instances, hypergraph colourings and colourings of bipartite graphs.Comment: 21 pages, revised version includes statement and proof of general
stopping times theorem (section 2.2), and additonal remarks in section
Non-Local Probes Do Not Help with Graph Problems
This work bridges the gap between distributed and centralised models of
computing in the context of sublinear-time graph algorithms. A priori, typical
centralised models of computing (e.g., parallel decision trees or centralised
local algorithms) seem to be much more powerful than distributed
message-passing algorithms: centralised algorithms can directly probe any part
of the input, while in distributed algorithms nodes can only communicate with
their immediate neighbours. We show that for a large class of graph problems,
this extra freedom does not help centralised algorithms at all: for example,
efficient stateless deterministic centralised local algorithms can be simulated
with efficient distributed message-passing algorithms. In particular, this
enables us to transfer existing lower bound results from distributed algorithms
to centralised local algorithms
Coupling with the stationary distribution and improved sampling for colorings and independent sets
We present an improved coupling technique for analyzing the mixing time of
Markov chains. Using our technique, we simplify and extend previous results for
sampling colorings and independent sets. Our approach uses properties of the
stationary distribution to avoid worst-case configurations which arise in the
traditional approach. As an application, we show that for ,
the Glauber dynamics on -colorings of a graph on vertices with maximum
degree converges in steps, assuming and that the graph is triangle-free. Previously, girth was needed.
As a second application, we give a polynomial-time algorithm for sampling
weighted independent sets from the Gibbs distribution of the hard-core lattice
gas model at fugacity , on a regular graph
on vertices of degree and girth . The best
known algorithm for general graphs currently assumes .Comment: Published at http://dx.doi.org/10.1214/105051606000000330 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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