19 research outputs found
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
Counting hypergraph matchings up to uniqueness threshold
We study the problem of approximately counting matchings in hypergraphs of
bounded maximum degree and maximum size of hyperedges. With an activity
parameter , each matching is assigned a weight .
The counting problem is formulated as computing a partition function that gives
the sum of the weights of all matchings in a hypergraph. This problem unifies
two extensively studied statistical physics models in approximate counting: the
hardcore model (graph independent sets) and the monomer-dimer model (graph
matchings).
For this model, the critical activity
is the threshold for the uniqueness of Gibbs measures on the infinite
-uniform -regular hypertree. Consider hypergraphs of maximum
degree at most and maximum size of hyperedges at most . We show that
when , there is an FPTAS for computing the partition
function; and when , there is a PTAS for computing the
log-partition function. These algorithms are based on the decay of correlation
(strong spatial mixing) property of Gibbs distributions. When , there is no PRAS for the partition function or the log-partition
function unless NPRP.
Towards obtaining a sharp transition of computational complexity of
approximate counting, we study the local convergence from a sequence of finite
hypergraphs to the infinite lattice with specified symmetry. We show a
surprising connection between the local convergence and the reversibility of a
natural random walk. This leads us to a barrier for the hardness result: The
non-uniqueness of infinite Gibbs measure is not realizable by any finite
gadgets
The Ising Partition Function: Zeros and Deterministic Approximation
We study the problem of approximating the partition function of the
ferromagnetic Ising model in graphs and hypergraphs. Our first result is a
deterministic approximation scheme (an FPTAS) for the partition function in
bounded degree graphs that is valid over the entire range of parameters
(the interaction) and (the external field), except for the case
(the "zero-field" case). A randomized algorithm (FPRAS)
for all graphs, and all , has long been known. Unlike most other
deterministic approximation algorithms for problems in statistical physics and
counting, our algorithm does not rely on the "decay of correlations" property.
Rather, we exploit and extend machinery developed recently by Barvinok, and
Patel and Regts, based on the location of the complex zeros of the partition
function, which can be seen as an algorithmic realization of the classical
Lee-Yang approach to phase transitions. Our approach extends to the more
general setting of the Ising model on hypergraphs of bounded degree and edge
size, where no previous algorithms (even randomized) were known for a wide
range of parameters. In order to achieve this extension, we establish a tight
version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a
classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results
on optimality of univariate Lee-Yang theorem
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Approximate counting, phase transitions and geometry of polynomials
In classical statistical physics, a phase transition is understood by studying the geometry (the zero-set) of an associated polynomial (the partition function). In this thesis, we will show that one can exploit this notion of phase transitions algorithmically, and conversely exploit the analysis of algorithms to understand phase transitions. As applications, we give efficient deterministic approximation algorithms (FPTAS) for counting -colorings, and for computing the partition function of the Ising model
Rapid mixing of hypergraph independent sets
We prove that the mixing time of the Glauber dynamics for sampling independent sets on n-vertex k-uniform hypergraphs is 0(n log n) when the maximum degree Δ satisfies Δ ≤ c2k/2, improving on the previous bound Bordewich and co-workers of Δ ≤ k − 2. This result brings the algorithmic bound to within a constant factor of the hardness bound of Bezakova and co-workers which showed that it is NP-hard to approximately count independent sets on hypergraphs when Δ ≥ 5·2k/2.Financial support by the EPSRC grant EP/L018896/1 (J.H.)
Approximation via Correlation Decay when Strong Spatial Mixing Fails
Approximate counting via correlation decay is the core algorithmic technique used in the sharp delineation of the computational phase transition that arises in the approximation of the partition function of antiferromagnetic 2-spin models. Previous analyses of correlation-decay algorithms implicitly depended on the occurrence of strong spatial mixing. This, roughly, means that one uses worst-case analysis of the recursive procedure that creates the subinstances. In this paper, we develop a new analysis method that is more refined than the worst-case analysis. We take the shape of instances in the computation tree into consideration and we amortize against certain “bad” instances that are created as the recursion proceeds. This enables us to show correlation decay and to obtain a fully polynomial-time approximation scheme (FPTAS) even when strong spatial mixing fails. We apply our technique to the problem of approximately counting independent sets in hypergraphs with degree upper bound and with a lower bound on the arity of hyperedges. Liu and Lin gave an FPTAS for and (lack of strong spatial mixing was the obstacle preventing this algorithm from being generalized to ). Our technique gives a tight result for , showing that there is an FPTAS for and . The best previously known approximation scheme for is the Markov-chain simulation based fully polynomial-time randomized approximation scheme (FPRAS) of Bordewich, Dyer, and Karpinski, which only works for . Our technique also applies for larger values of , giving an FPTAS for . This bound is not substantially stronger than existing randomized results in the literature. Nevertheless, it gives the first deterministic approximation scheme in this regime. Moreover, unlike existing results, it leads to an FPTAS for counting dominating sets in regular graphs with sufficiently large degree. We further demonstrate that in the hypergraph independent set model, approximating the partition function is NP-hard even within the uniqueness regime. Also, approximately counting dominating sets of bounded-degree graphs (without the regularity restriction) is NP-hard
Perfect sampling from spatial mixing
We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth. In particular, in graphs with subexponential neighborhood growth like [Formula: see text] , our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing. As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer‐dimer models in such graphs
Approximate Counting, the Lovasz Local Lemma and Inference in Graphical Models
In this paper we introduce a new approach for approximately counting in
bounded degree systems with higher-order constraints. Our main result is an
algorithm to approximately count the number of solutions to a CNF formula
when the width is logarithmic in the maximum degree. This closes an
exponential gap between the known upper and lower bounds.
Moreover our algorithm extends straightforwardly to approximate sampling,
which shows that under Lov\'asz Local Lemma-like conditions it is not only
possible to find a satisfying assignment, it is also possible to generate one
approximately uniformly at random from the set of all satisfying assignments.
Our approach is a significant departure from earlier techniques in approximate
counting, and is based on a framework to bootstrap an oracle for computing
marginal probabilities on individual variables. Finally, we give an application
of our results to show that it is algorithmically possible to sample from the
posterior distribution in an interesting class of graphical models.Comment: 25 pages, 2 figure
Optimal zero-free regions for the independence polynomial of bounded degree hypergraphs
In this paper we investigate the distribution of zeros of the independence
polynomial of hypergraphs of maximum degree . For graphs the largest
zero-free disk around zero was described by Shearer as having radius
. Recently it was shown
by Galvin et al. that for hypergraphs the disk of radius
is zero-free; however, it was conjectured that the actual truth should be
. We show that this is indeed the case. We also show that
there exists an open region around the interval
that is zero-free for hypergraphs
of maximum degree , which extends the result of Peters and Regts from
graphs to hypergraphs. Finally, we determine the radius of the largest
zero-free disk for the family of bounded degree -uniform linear hypertrees
in terms of and .Comment: 34 pages, 4 figure