46 research outputs found
A graph polynomial for independent sets of bipartite graphs
We introduce a new graph polynomial that encodes interesting properties of
graphs, for example, the number of matchings and the number of perfect
matchings. Most importantly, for bipartite graphs the polynomial encodes the
number of independent sets (#BIS).
We analyze the complexity of exact evaluation of the polynomial at rational
points and show that for most points exact evaluation is #P-hard (assuming the
generalized Riemann hypothesis) and for the rest of the points exact evaluation
is trivial.
We conjecture that a natural Markov chain can be used to approximately
evaluate the polynomial for a range of parameters. The conjecture, if true,
would imply an approximate counting algorithm for #BIS, a problem shown, by
[Dyer et al. 2004], to be complete (with respect to, so called, AP-reductions)
for a rich logically defined sub-class of #P. We give a mild support for our
conjecture by proving that the Markov chain is rapidly mixing on trees. As a
by-product we show that the "single bond flip" Markov chain for the random
cluster model is rapidly mixing on constant tree-width graphs
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
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