19 research outputs found
Approximate Capacities of Two-Dimensional Codes by Spatial Mixing
We apply several state-of-the-art techniques developed in recent advances of
counting algorithms and statistical physics to study the spatial mixing
property of the two-dimensional codes arising from local hard (independent set)
constraints, including: hard-square, hard-hexagon, read/write isolated memory
(RWIM), and non-attacking kings (NAK). For these constraints, the strong
spatial mixing would imply the existence of polynomial-time approximation
scheme (PTAS) for computing the capacity. It was previously known for the
hard-square constraint the existence of strong spatial mixing and PTAS. We show
the existence of strong spatial mixing for hard-hexagon and RWIM constraints by
establishing the strong spatial mixing along self-avoiding walks, and
consequently we give PTAS for computing the capacities of these codes. We also
show that for the NAK constraint, the strong spatial mixing does not hold along
self-avoiding walks
FPTAS for Hardcore and Ising Models on Hypergraphs
Hardcore and Ising models are two most important families of two state spin
systems in statistic physics. Partition function of spin systems is the center
concept in statistic physics which connects microscopic particles and their
interactions with their macroscopic and statistical properties of materials
such as energy, entropy, ferromagnetism, etc. If each local interaction of the
system involves only two particles, the system can be described by a graph. In
this case, fully polynomial-time approximation scheme (FPTAS) for computing the
partition function of both hardcore and anti-ferromagnetic Ising model was
designed up to the uniqueness condition of the system. These result are the
best possible since approximately computing the partition function beyond this
threshold is NP-hard. In this paper, we generalize these results to general
physics systems, where each local interaction may involves multiple particles.
Such systems are described by hypergraphs. For hardcore model, we also provide
FPTAS up to the uniqueness condition, and for anti-ferromagnetic Ising model,
we obtain FPTAS where a slightly stronger condition holds
Improved Bounds on the Phase Transition for the Hard-Core Model in 2-Dimensions
For the hard-core lattice gas model defined on independent sets weighted by
an activity , we study the critical activity
for the uniqueness/non-uniqueness threshold on the 2-dimensional integer
lattice . The conjectured value of the critical activity is
approximately . Until recently, the best lower bound followed from
algorithmic results of Weitz (2006). Weitz presented an FPTAS for approximating
the partition function for graphs of constant maximum degree when
where is the
infinite, regular tree of degree . His result established a certain
decay of correlations property called strong spatial mixing (SSM) on
by proving that SSM holds on its self-avoiding walk tree
where and is an ordering on the neighbors of vertex . As
a consequence he obtained that . Restrepo et al. (2011) improved Weitz's approach for
the particular case of and obtained that
. In this paper, we establish an upper bound for
this approach, by showing that, for all , SSM does not hold on
when . We also present a
refinement of the approach of Restrepo et al. which improves the lower bound to
.Comment: 19 pages, 1 figure. Polished proofs and examples compared to earlier
versio
FPTAS for #BIS with Degree Bounds on One Side
Counting the number of independent sets for a bipartite graph (#BIS) plays a
crucial role in the study of approximate counting. It has been conjectured that
there is no fully polynomial-time (randomized) approximation scheme
(FPTAS/FPRAS) for #BIS, and it was proved that the problem for instances with a
maximum degree of is already as hard as the general problem. In this paper,
we obtain a surprising tractability result for a family of #BIS instances. We
design a very simple deterministic fully polynomial-time approximation scheme
(FPTAS) for #BIS when the maximum degree for one side is no larger than .
There is no restriction for the degrees on the other side, which do not even
have to be bounded by a constant. Previously, FPTAS was only known for
instances with a maximum degree of for both sides.Comment: 15 pages, to appear in STOC 2015; Improved presentations from
previous version
Correlation Decay up to Uniqueness in Spin Systems
We give a complete characterization of the two-state anti-ferromagnetic spin
systems which are of strong spatial mixing on general graphs. We show that a
two-state anti-ferromagnetic spin system is of strong spatial mixing on all
graphs of maximum degree at most \Delta if and only if the system has a unique
Gibbs measure on infinite regular trees of degree up to \Delta, where \Delta
can be either bounded or unbounded. As a consequence, there exists an FPTAS for
the partition function of a two-state anti-ferromagnetic spin system on graphs
of maximum degree at most \Delta when the uniqueness condition is satisfied on
infinite regular trees of degree up to \Delta. In particular, an FPTAS exists
for arbitrary graphs if the uniqueness is satisfied on all infinite regular
trees. This covers as special cases all previous algorithmic results for
two-state anti-ferromagnetic systems on general-structure graphs.
Combining with the FPRAS for two-state ferromagnetic spin systems of
Jerrum-Sinclair and Goldberg-Jerrum-Paterson, and the very recent hardness
results of Sly-Sun and independently of Galanis-Stefankovic-Vigoda, this gives
a complete classification, except at the phase transition boundary, of the
approximability of all two-state spin systems, on either degree-bounded
families of graphs or family of all graphs.Comment: 27 pages, submitted for publicatio
Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs
In a seminal paper [10], Weitz gave a deterministic fully polynomial approximation scheme for counting exponentially weighted independent sets (which is the same as approximating the partition function of the hard-core model from statistical physics) in graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the in nite d-regular tree. More recently Sly [8] (see also [1]) showed that this is optimal in the sense that if there is an FPRAS for the hard-core partition function on graphs of maximum degree d for activities larger than the critical activity on the in nite d-regular tree then NP = RP. In this paper we extend Weitz's approach to derive a deterministic fully polynomial approximation scheme for the partition function of general two-state anti-ferromagnetic spin systems on graphs of maximum degree d, up to the corresponding critical point on the d-regular tree. The main ingredient of our result is a proof that for two-state anti-ferromagnetic spin systems on the d-regular tree, weak spatial mixing implies strong spatial mixing. This in turn uses a message-decay argument which extends a similar approach proposed recently for the hard-core model by Restrepo et al [7] to the case of general two-state anti-ferromagnetic spin systems
Spatial mixing and approximation algorithms for graphs with bounded connective constant
The hard core model in statistical physics is a probability distribution on
independent sets in a graph in which the weight of any independent set I is
proportional to lambda^(|I|), where lambda > 0 is the vertex activity. We show
that there is an intimate connection between the connective constant of a graph
and the phenomenon of strong spatial mixing (decay of correlations) for the
hard core model; specifically, we prove that the hard core model with vertex
activity lambda < lambda_c(Delta + 1) exhibits strong spatial mixing on any
graph of connective constant Delta, irrespective of its maximum degree, and
hence derive an FPTAS for the partition function of the hard core model on such
graphs. Here lambda_c(d) := d^d/(d-1)^(d+1) is the critical activity for the
uniqueness of the Gibbs measure of the hard core model on the infinite d-ary
tree. As an application, we show that the partition function can be efficiently
approximated with high probability on graphs drawn from the random graph model
G(n,d/n) for all lambda < e/d, even though the maximum degree of such graphs is
unbounded with high probability.
We also improve upon Weitz's bounds for strong spatial mixing on bounded
degree graphs (Weitz, 2006) by providing a computationally simple method which
uses known estimates of the connective constant of a lattice to obtain bounds
on the vertex activities lambda for which the hard core model on the lattice
exhibits strong spatial mixing. Using this framework, we improve upon these
bounds for several lattices including the Cartesian lattice in dimensions 3 and
higher.
Our techniques also allow us to relate the threshold for the uniqueness of
the Gibbs measure on a general tree to its branching factor (Lyons, 1989).Comment: 26 pages. In October 2014, this paper was superseded by
arxiv:1410.2595. Before that, an extended abstract of this paper appeared in
Proc. IEEE Symposium on the Foundations of Computer Science (FOCS), 2013, pp.
300-30