15,844 research outputs found
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 âbadâ 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
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 anti-ferromagnetic two-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 sub-instances. 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 amortise against certain âbadâ instances that are created as the recursion proceeds. This enables us to show correlation decay and to obtain an 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 â and with a lower bound k on the arity of hyperedges. Liu and Lin gave an FPTAS for k â„ 2 and â †5 (lack of strong spatial mixing was the obstacle preventing this algorithm from being generalised to â = 6). Our technique gives a tight result for â = 6, showing that there is an FPTAS for k â„ 3 and â †6. The best previously-known approximation scheme for â = 6 is the Markov-chain simulation based FPRAS of Bordewich, Dyer and Karpinski, which only works for k â„ 8. Our technique also applies for larger values of k, giving an FPTAS for k â„ 1.66â. This bound is not as strong as existing randomised results, for technical reasons that are discussed in the paper. Nevertheless, it gives the first deterministic approximation schemes in this regime. We further demonstrate that in the hypergraph independent set model, approximating the partition function is NP-hard even within the uniqueness regime.</p
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
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
Fluidization of collisionless plasma turbulence
In a collisionless, magnetized plasma, particles may stream freely along
magnetic-field lines, leading to phase "mixing" of their distribution function
and consequently to smoothing out of any "compressive" fluctuations (of
density, pressure, etc.,). This rapid mixing underlies Landau damping of these
fluctuations in a quiescent plasma-one of the most fundamental physical
phenomena that make plasma different from a conventional fluid. Nevertheless,
broad power-law spectra of compressive fluctuations are observed in turbulent
astrophysical plasmas (most vividly, in the solar wind) under conditions
conducive to strong Landau damping. Elsewhere in nature, such spectra are
normally associated with fluid turbulence, where energy cannot be dissipated in
the inertial scale range and is therefore cascaded from large scales to small.
By direct numerical simulations and theoretical arguments, it is shown here
that turbulence of compressive fluctuations in collisionless plasmas strongly
resembles one in a collisional fluid and does have broad power-law spectra.
This "fluidization" of collisionless plasmas occurs because phase mixing is
strongly suppressed on average by "stochastic echoes", arising due to nonlinear
advection of the particle distribution by turbulent motions. Besides resolving
the long-standing puzzle of observed compressive fluctuations in the solar
wind, our results suggest a conceptual shift for understanding kinetic plasma
turbulence generally: rather than being a system where Landau damping plays the
role of dissipation, a collisionless plasma is effectively dissipationless
except at very small scales. The universality of "fluid" turbulence physics is
thus reaffirmed even for a kinetic, collisionless system
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
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