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18
A PolynomialTime Approximation Algorithm for the Permanent of a Matrix with NonNegative Entries
 Journal of the ACM
, 2004
"... Abstract. We present a polynomialtime randomized algorithm for estimating the permanent of an arbitrary n ×n matrix with nonnegative entries. This algorithm—technically a “fullypolynomial randomized approximation scheme”—computes an approximation that is, with high probability, within arbitrarily ..."
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Cited by 324 (25 self)
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Abstract. We present a polynomialtime randomized algorithm for estimating the permanent of an arbitrary n ×n matrix with nonnegative entries. This algorithm—technically a “fullypolynomial randomized approximation scheme”—computes an approximation that is, with high probability, within arbitrarily small specified relative error of the true value of the permanent. Categories and Subject Descriptors: F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical
The Markov Chain Monte Carlo method: an approach to approximate counting and integration
, 1996
"... In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stocha ..."
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Cited by 234 (13 self)
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In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stochastic processes hardly touches on the sort of nonasymptotic analysis required in this application. As a consequence, it had previously not been possible to make useful, mathematically rigorous statements about the quality of the estimates obtained. Within the last ten years, analytical tools have been devised with the aim of correcting this deficiency. As well as permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics, the introduction of these tools has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization. The “Markov chain Monte Carlo ” method has been applied to a variety of such problems, and often provides the only known efficient (i.e., polynomial time) solution technique.
Analyzing Glauber Dynamics by Comparison of Markov Chains
 Journal of Mathematical Physics
, 1999
"... A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or "Glauber dynamics." Typically these single ..."
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Cited by 64 (12 self)
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A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or "Glauber dynamics." Typically these single site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems. In this work we use the comparison technique of Diaconis and SaloffCoste to show that several of the natural single point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed. This allows us to give polynomial bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of c...
The SwendsenWang process does not always mix rapidly
 Proc. 29th ACM Symp. on Theory of Computing
, 1997
"... The SwendsenWang process provides one possible dynamics for the Qstate Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The l ..."
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Cited by 39 (2 self)
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The SwendsenWang process provides one possible dynamics for the Qstate Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The legitimacy of such simulations depends on the rate of convergence of the process to equilibrium, often known as the mixing rate. Empirical observations suggest that the SwendsenWang process mixes rapidly in many instances of practical interest. In spite of this, we show that there are occasions on which the SwendsenWang process requires exponential time (in the size of the system) to approach equilibrium.
Statistical Mechanics, ThreeDimensionality and NPcompleteness I. Universality of Intractability for the Partition Function of the Ising Model Across NonPlanar Lattices (Extended Abstract)
"... This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NPcomplete. We provide rigorous solutions to several working ..."
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Cited by 28 (1 self)
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This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NPcomplete. We provide rigorous solutions to several working conjectures in the statistical mechanics literature, such as the CrossedBonds conjecture, and the impossibility to compute effectively the partition functions for any threedimensional lattice Ising model � these conjectures apply to the Onsager algebraic method, the Fermion operators method, and the combinatorial method based on Pfaffians. The fundamental results of the area, including those of Onsager, Kac, Feynman, Fisher, Kasteleyn, Temperley, Green, Hurst and more recently Barahona: for every Planar crystal lattice the partition functions for the nite sublattices can be computed in polynomialtime, paired with the results of this paper: for every NonPlanar crystal lattice computing the parition functions for the finite sublattices is NPcomplete, provide an exact characterization for several of the most studied Ising models. Our results settle at once, for several models, (1) the 2D nonplanar vs. 2D planar, (2) the nextnearest neighbour
Computational Complexity of Loss Networks
 THEORETICAL COMPUTER SCIENCE
, 1993
"... In this paper we examine the theoretical limits on developing algorithms to find blocking probabilities in a general loss network. We demonstrate that exactly computing the blocking probabilities of a loss network is a #Pcomplete problem. We also show that a general algorithm for approximating the ..."
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Cited by 25 (2 self)
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In this paper we examine the theoretical limits on developing algorithms to find blocking probabilities in a general loss network. We demonstrate that exactly computing the blocking probabilities of a loss network is a #Pcomplete problem. We also show that a general algorithm for approximating the blocking probabilities is also intractable unless RP=NP, which seems unlikely according to current common notions in complexity theory. Given these results, we examine implications for designing practical algorithms for finding blocking probabilities in special cases.
Approximating the Number of MonomerDimer Coverings of a Lattice
 Journal of Statistical Physics
, 1996
"... The paper studies the problem of counting the number of coverings of a ddimensional rectangular lattice by a specified number of monomers and dimers. This problem arises in several models in statistical physics, and has been widely studied. A classical technique due to Fisher, Kasteleyn and Temper ..."
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Cited by 19 (2 self)
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The paper studies the problem of counting the number of coverings of a ddimensional rectangular lattice by a specified number of monomers and dimers. This problem arises in several models in statistical physics, and has been widely studied. A classical technique due to Fisher, Kasteleyn and Temperley solves the problem exactly in two dimensions when the number of monomers is zero (the dimer covering problem), but is not applicable in higher dimensions or in the presence of monomers. This paper presents the first provably polynomial time approximation algorithms for computing the number of coverings with any specified number of monomers in ddimensional rectangular lattices with periodic boundaries, for any fixed dimension d , and in twodimensional lattices with fixed boundaries. The algorithms are based on Monte Carlo simulation of a suitable Markov chain, and, in contrast to most Monte Carlo algorithms in statistical physics, have rigorously derived performance guarantees that do n...
The Computational Complexity of Generating Random Fractals
 J. Stat. Phys
, 1996
"... In this paper we examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion limited aggregation and several widely used algorithms for equilibrating the Ising model ar ..."
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Cited by 16 (6 self)
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In this paper we examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion limited aggregation and several widely used algorithms for equilibrating the Ising model are shown to be highly sequential; it is unlikely they can be simulated efficiently in parallel. This is in contrast to Mandelbrot percolation that can be simulated in constant parallel time. Our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory. In addition, the results may serve as a guide to simulation physics. Keywords: Cluster algorithms, computational complexity, diffusion limited aggregation, Ising model, Metropolis algorithm, Pcompleteness 1
Matchings in Lattice Graphs
 Proceedings of the 25th ACM Symposium on Theory of Computing
, 1993
"... We study the problem of counting the number of matchings of given cardinality in a ddimensional rectangular lattice. This problem arises in several models in statistical physics, including monomerdimer systems and cellcluster theory. A classical algorithm due to Fisher, Kasteleyn and Temperley c ..."
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Cited by 7 (4 self)
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We study the problem of counting the number of matchings of given cardinality in a ddimensional rectangular lattice. This problem arises in several models in statistical physics, including monomerdimer systems and cellcluster theory. A classical algorithm due to Fisher, Kasteleyn and Temperley counts perfect matchings exactly in two dimensions, but is not applicable in higher dimensions and does not allow one to count matchings of arbitrary cardinality. In this paper, we present the first efficient approximation algorithms for counting matchings of arbitrary cardinality in (i) ddimensional "periodic" lattices (i.e., with wraparound edges) in any fixed dimension d; and (ii) twodimensional lattices with "fixed boundary conditions" (i.e., no wraparound edges). Our technique generalizes to approximately counting matchings in any bipartite graph that is the Cayley graph of some finite group. y CNRS, Ecole Normale Sup'erieure de Lyon, France. Part of this work was done while this au...