100 research outputs found
The complexity of approximately counting in 2-spin systems on -uniform bounded-degree hypergraphs
One of the most important recent developments in the complexity of
approximate counting is the classification of the complexity of approximating
the partition functions of antiferromagnetic 2-spin systems on bounded-degree
graphs. This classification is based on a beautiful connection to the so-called
uniqueness phase transition from statistical physics on the infinite
-regular tree. Our objective is to study the impact of this
classification on unweighted 2-spin models on -uniform hypergraphs. As has
already been indicated by Yin and Zhao, the connection between the uniqueness
phase transition and the complexity of approximate counting breaks down in the
hypergraph setting. Nevertheless, we show that for every non-trivial symmetric
-ary Boolean function there exists a degree bound so that for
all the following problem is NP-hard: given a
-uniform hypergraph with maximum degree at most , approximate the
partition function of the hypergraph 2-spin model associated with . It is
NP-hard to approximate this partition function even within an exponential
factor. By contrast, if is a trivial symmetric Boolean function (e.g., any
function that is excluded from our result), then the partition function of
the corresponding hypergraph 2-spin model can be computed exactly in polynomial
time
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
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 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
Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results
Recent results establish for 2-spin antiferromagnetic systems that the
computational complexity of approximating the partition function on graphs of
maximum degree D undergoes a phase transition that coincides with the
uniqueness phase transition on the infinite D-regular tree. For the
ferromagnetic Potts model we investigate whether analogous hardness results
hold. Goldberg and Jerrum showed that approximating the partition function of
the ferromagnetic Potts model is at least as hard as approximating the number
of independent sets in bipartite graphs (#BIS-hardness). We improve this
hardness result by establishing it for bipartite graphs of maximum degree D. We
first present a detailed picture for the phase diagram for the infinite
D-regular tree, giving a refined picture of its first-order phase transition
and establishing the critical temperature for the coexistence of the disordered
and ordered phases. We then prove for all temperatures below this critical
temperature that it is #BIS-hard to approximate the partition function on
bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to
approximate the number of k-colorings on bipartite graphs of maximum degree D
when k <= D/(2 ln D).
The #BIS-hardness result for the ferromagnetic Potts model uses random
bipartite regular graphs as a gadget in the reduction. The analysis of these
random graphs relies on recent connections between the maxima of the
expectation of their partition function, attractive fixpoints of the associated
tree recursions, and induced matrix norms. We extend these connections to
random regular graphs for all ferromagnetic models and establish the Bethe
prediction for every ferromagnetic spin system on random regular graphs. We
also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm
is torpidly mixing on random D-regular graphs at the critical temperature for
large q.Comment: To appear in SIAM J. Computin
The complexity of approximating the matching polynomial in the complex plane
We study the problem of approximating the value of the matching polynomial on
graphs with edge parameter , where takes arbitrary values in
the complex plane.
When is a positive real, Jerrum and Sinclair showed that the problem
admits an FPRAS on general graphs. For general complex values of ,
Patel and Regts, building on methods developed by Barvinok, showed that the
problem admits an FPTAS on graphs of maximum degree as long as
is not a negative real number less than or equal to
. Our first main result completes the picture for the
approximability of the matching polynomial on bounded degree graphs. We show
that for all and all real less than ,
the problem of approximating the value of the matching polynomial on graphs of
maximum degree with edge parameter is #P-hard.
We then explore whether the maximum degree parameter can be replaced by the
connective constant. Sinclair et al. showed that for positive real it
is possible to approximate the value of the matching polynomial using a
correlation decay algorithm on graphs with bounded connective constant (and
potentially unbounded maximum degree). We first show that this result does not
extend in general in the complex plane; in particular, the problem is #P-hard
on graphs with bounded connective constant for a dense set of values
on the negative real axis. Nevertheless, we show that the result does extend
for any complex value that does not lie on the negative real axis. Our
analysis accounts for complex values of using geodesic distances in
the complex plane in the metric defined by an appropriate density function
Inapproximability of the independent set polynomial in the complex plane
We study the complexity of approximating the independent set polynomial
of a graph with maximum degree when the activity
is a complex number.
This problem is already well understood when is real using
connections to the -regular tree . The key concept in that case is
the "occupation ratio" of the tree . This ratio is the contribution to
from independent sets containing the root of the tree, divided
by itself. If is such that the occupation ratio
converges to a limit, as the height of grows, then there is an FPTAS for
approximating on a graph with maximum degree .
Otherwise, the approximation problem is NP-hard.
Unsurprisingly, the case where is complex is more challenging.
Peters and Regts identified the complex values of for which the
occupation ratio of the -regular tree converges. These values carve a
cardioid-shaped region in the complex plane. Motivated by the
picture in the real case, they asked whether marks the true
approximability threshold for general complex values .
Our main result shows that for every outside of ,
the problem of approximating on graphs with maximum degree
at most is indeed NP-hard. In fact, when is outside of
and is not a positive real number, we give the stronger result
that approximating is actually #P-hard. If is a
negative real number outside of , we show that it is #P-hard to
even decide whether , resolving in the affirmative a conjecture
of Harvey, Srivastava and Vondrak.
Our proof techniques are based around tools from complex analysis -
specifically the study of iterative multivariate rational maps
Swendsen-Wang Algorithm on the Mean-Field Potts Model
We study the q-state ferromagnetic Potts model on the n-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. The case q=2 (the Swendsen-Wang algorithm for the ferromagnetic Ising model) undergoes a slow-down at the uniqueness/non-uniqueness critical temperature for the infinite Delta-regular tree (Long et al., 2014) but yet still has polynomial mixing time at all (inverse) temperatures beta>0 (Cooper et al., 2000). In contrast for q>=3 there are two critical temperatures 0=beta_rc. These results complement refined results of Cuff et al. (2012) on the mixing time of the Glauber dynamics for the ferromagnetic Potts model. The most interesting aspect of our analysis is at the critical temperature beta=beta_u, which requires a delicate choice of a potential function to balance the conflating factors for the slow drift away from a fixed point (which is repulsive but not Jacobian repulsive): close to the fixed point the variance from the percolation step dominates and sufficiently far from the fixed point the dynamics of the size of the dominant color class takes over
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