112 research outputs found
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
Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models
Recent inapproximability results of Sly (2010), together with an
approximation algorithm presented by Weitz (2006) establish a beautiful picture
for the computational complexity of approximating the partition function of the
hard-core model. Let denote the critical activity for the
hard-model on the infinite -regular tree. Weitz presented an FPTAS for
the partition function when for graphs with
constant maximum degree . In contrast, Sly showed that for all
, there exists such that (unless RP=NP) there
is no FPRAS for approximating the partition function on graphs of maximum
degree for activities satisfying
.
We prove that a similar phenomenon holds for the antiferromagnetic Ising
model. Recent results of Li et al. and Sinclair et al. extend Weitz's approach
to any 2-spin model, which includes the antiferromagnetic Ising model, to yield
an FPTAS for the partition function for all graphs of constant maximum degree
when the parameters of the model lie in the uniqueness regime of the
infinite tree . We prove the complementary result that for the
antiferrogmanetic Ising model without external field that, unless RP=NP, for
all , there is no FPRAS for approximating the partition function
on graphs of maximum degree when the inverse temperature lies in the
non-uniqueness regime of the infinite tree . Our results extend to a
region of the parameter space for general 2-spin models. Our proof works by
relating certain second moment calculations for random -regular
bipartite graphs to the tree recursions used to establish the critical points
on the infinite tree.Comment: Journal version (no changes
Swendsen-Wang Algorithm on the Mean-Field Potts Model
We study the -state ferromagnetic Potts model on the -vertex complete
graph known as the mean-field (Curie-Weiss) model. We analyze the Swendsen-Wang
algorithm which is a Markov chain that utilizes the random cluster
representation for the ferromagnetic Potts model to recolor large sets of
vertices in one step and potentially overcomes obstacles that inhibit
single-site Glauber dynamics. Long et al. studied the case , the
Swendsen-Wang algorithm for the mean-field ferromagnetic Ising model, and
showed that the mixing time satisfies: (i) for ,
(ii) for , (iii) for
, where is the critical temperature for the
ordered/disordered phase transition. In contrast, for there are two
critical temperatures that are relevant. We prove that
the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts
model on the -vertex complete graph satisfies: (i) for
, (ii) for , (iii)
for , and (iv)
for . These results complement refined
results of Cuff et al. on the mixing time of the Glauber dynamics for the
ferromagnetic Potts model.Comment: To appear in Random Structures & Algorithm
Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions
The random-cluster (FK) model is a key tool for the study of phase transitions and for the design of efficient Markov chain Monte Carlo (MCMC) sampling algorithms for the Ising/Potts model. It is well-known that in the high-temperature region beta<beta_c(q) of the q-state Ising/Potts model on an n x n box Lambda_n of the integer lattice Z^2, spin correlations decay exponentially fast; this property holds even arbitrarily close to the boundary of Lambda_n and uniformly over all boundary conditions. A direct consequence of this property is that the corresponding single-site update Markov chain, known as the Glauber dynamics, mixes in optimal O(n^2 log{n}) steps on Lambda_{n} for all choices of boundary conditions. We study the effect of boundary conditions on the FK-dynamics, the analogous Glauber dynamics for the random-cluster model.
On Lambda_n the random-cluster model with parameters (p,q) has a sharp phase transition at p = p_c(q). Unlike the Ising/Potts model, the random-cluster model has non-local interactions which can be forced by boundary conditions: external wirings of boundary vertices of Lambda_n. We consider the broad and natural class of boundary conditions that are realizable as a configuration on Z^2 Lambda_n. Such boundary conditions can have many macroscopic wirings and impose long-range correlations even at very high temperatures (p 1 and p != p_c(q) the mixing time of the FK-dynamics is polynomial in n for every realizable boundary condition. Previously, for boundary conditions that do not carry long-range information (namely wired and free), Blanca and Sinclair (2017) had proved that the FK-dynamics in the same setting mixes in optimal O(n^2 log n) time. To illustrate the difficulties introduced by general boundary conditions, we also construct a class of non-realizable boundary conditions that induce slow (stretched-exponential) convergence at high temperatures
Inapproximability for Antiferromagnetic Spin Systems in the Tree Non-Uniqueness Region
A remarkable connection has been established for antiferromagnetic 2-spin
systems, including the Ising and hard-core models, showing that the
computational complexity of approximating the partition function for graphs
with maximum degree D undergoes a phase transition that coincides with the
statistical physics uniqueness/non-uniqueness phase transition on the infinite
D-regular tree. Despite this clear picture for 2-spin systems, there is little
known for multi-spin systems. We present the first analog of the above
inapproximability results for multi-spin systems.
The main difficulty in previous inapproximability results was analyzing the
behavior of the model on random D-regular bipartite graphs, which served as the
gadget in the reduction. To this end one needs to understand the moments of the
partition function. Our key contribution is connecting: (i) induced matrix
norms, (ii) maxima of the expectation of the partition function, and (iii)
attractive fixed points of the associated tree recursions (belief propagation).
The view through matrix norms allows a simple and generic analysis of the
second moment for any spin system on random D-regular bipartite graphs. This
yields concentration results for any spin system in which one can analyze the
maxima of the first moment. The connection to fixed points of the tree
recursions enables an analysis of the maxima of the first moment for specific
models of interest.
For k-colorings we prove that for even k, in the tree non-uniqueness region
(which corresponds to k<D) it is NP-hard, unless NP=RP, to approximate the
number of colorings for triangle-free D-regular graphs. Our proof extends to
the antiferromagnetic Potts model, and, in fact, to every antiferromagnetic
model under a mild condition
Analysis of top-swap shuffling for genome rearrangements
We study Markov chains which model genome rearrangements. These models are
useful for studying the equilibrium distribution of chromosomal lengths, and
are used in methods for estimating genomic distances. The primary Markov chain
studied in this paper is the top-swap Markov chain. The top-swap chain is a
card-shuffling process with cards divided over decks, where the cards
are ordered within each deck. A transition consists of choosing a random pair
of cards, and if the cards lie in different decks, we cut each deck at the
chosen card and exchange the tops of the two decks. We prove precise bounds on
the relaxation time (inverse spectral gap) of the top-swap chain. In
particular, we prove the relaxation time is . This resolves an
open question of Durrett.Comment: Published in at http://dx.doi.org/10.1214/105051607000000177 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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