1,332 research outputs found
The complexity of approximating the complex-valued Potts model
We study the complexity of approximating the partition function of the
-state Potts model and the closely related Tutte polynomial for complex
values of the underlying parameters. Apart from the classical connections with
quantum computing and phase transitions in statistical physics, recent work in
approximate counting has shown that the behaviour in the complex plane, and
more precisely the location of zeros, is strongly connected with the complexity
of the approximation problem, even for positive real-valued parameters.
Previous work in the complex plane by Goldberg and Guo focused on , which
corresponds to the case of the Ising model; for , the behaviour in the
complex plane is not as well understood and most work applies only to the
real-valued Tutte plane.
Our main result is a complete classification of the complexity of the
approximation problems for all non-real values of the parameters, by
establishing \#P-hardness results that apply even when restricted to planar
graphs. Our techniques apply to all and further complement/refine
previous results both for the Ising model and the Tutte plane, answering in
particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in
the context of quantum computations.Comment: 58 pages. Changes on version 2: minor change
Approximate counting via complex zero-free regions and spectral independence
This thesis investigates fundamental problems in approximate counting that arise in the field of statistical mechanics. Building upon recent advancements in the area, our research aims to enhance our understanding of the computational complexity of sampling from the Ising and Potts models, as well as the random -SAT model.
The -state Potts model is a spin model in which each particle is randomly assigned a spin (out of possible spins), where the probability of a certain assignment depends on how many adjacent particles present the same spin. The edge interaction of the model is a parameter that quantifies the strength of interaction between two adjacent particles. The Ising model corresponds to the Potts model with . Sampling from these models is inherently connected to approximating the partition function of the model, a graph polynomial that encodes several aggregate thermodynamic properties of the system. In addition to classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane of these partition functions, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Thus, following this trend in both statistical physics and algorithmic research, we allow the edge interaction to be any complex number.
First, we study the complexity of approximating the partition function of the -state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Previous work in the complex plane by Goldberg and Guo focused on ; for , the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing \#P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in the context of quantum computations.
Secondly, we investigate the complexity of approximating the partition function \ising(G; \beta) of the Ising model in terms of the relation between the edge interaction and a parameter which is an upper bound on the maximum degree of the input graph . In this thesis we establish both new tractability and inapproximability results. Our tractability results show that \ising(-; \beta) has an FPTAS when and . These are the first results to show intractability of approximating \ising(-, \beta) on bounded degree graphs with complex . Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model.
Finally, we exploit the recently successful framework of spectral independence to analyse the mixing time of a Markov chain, and we apply it in order to sample satisfying assignments of -CNF formulas. Our analysis leads to a nearly linear-time algorithm to approximately sample satisfying assignments in the random -SAT model when the density of the random formula scales exponentially with , where is the number of variables and is the number of clauses. The best previously known sampling algorithm for the random -SAT model applies when the density of the formula is less than and runs in time . Our algorithm achieves a significantly faster running time of and samples satisfying assignments up to density . The main challenge in our setting is the presence of many variables with unbounded degree, which causes significant correlations within the formula and impedes the application of relevant Markov chain methods from the bounded-degree setting
Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
In this paper we show a new way of constructing deterministic polynomial-time
approximation algorithms for computing complex-valued evaluations of a large
class of graph polynomials on bounded degree graphs. In particular, our
approach works for the Tutte polynomial and independence polynomial, as well as
partition functions of complex-valued spin and edge-coloring models.
More specifically, we define a large class of graph polynomials
and show that if and there is a disk centered at zero in the
complex plane such that does not vanish on for all bounded degree
graphs , then for each in the interior of there exists a
deterministic polynomial-time approximation algorithm for evaluating at
. This gives an explicit connection between absence of zeros of graph
polynomials and the existence of efficient approximation algorithms, allowing
us to show new relationships between well-known conjectures.
Our work builds on a recent line of work initiated by. Barvinok, which
provides a new algorithmic approach besides the existing Markov chain Monte
Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In
particular a tiny error in Proposition 4.4 has been fixed. The introduction
and concluding remarks have also been rewritten to incorporate the most
recent developments. Accepted for publication in SIAM Journal on Computatio
Neural Distributed Autoassociative Memories: A Survey
Introduction. Neural network models of autoassociative, distributed memory
allow storage and retrieval of many items (vectors) where the number of stored
items can exceed the vector dimension (the number of neurons in the network).
This opens the possibility of a sublinear time search (in the number of stored
items) for approximate nearest neighbors among vectors of high dimension. The
purpose of this paper is to review models of autoassociative, distributed
memory that can be naturally implemented by neural networks (mainly with local
learning rules and iterative dynamics based on information locally available to
neurons). Scope. The survey is focused mainly on the networks of Hopfield,
Willshaw and Potts, that have connections between pairs of neurons and operate
on sparse binary vectors. We discuss not only autoassociative memory, but also
the generalization properties of these networks. We also consider neural
networks with higher-order connections and networks with a bipartite graph
structure for non-binary data with linear constraints. Conclusions. In
conclusion we discuss the relations to similarity search, advantages and
drawbacks of these techniques, and topics for further research. An interesting
and still not completely resolved question is whether neural autoassociative
memories can search for approximate nearest neighbors faster than other index
structures for similarity search, in particular for the case of very high
dimensional vectors.Comment: 31 page
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