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28
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.
A Deterministic Strongly Polynomial Algorithm for Matrix Scaling and Approximate Permanents
"... We present a deterministic strongly polynomial algorithm that computes the permanent of a nonnegative n x n matrix to within a multiplicative factor of e^n. To this end ..."
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Cited by 63 (8 self)
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We present a deterministic strongly polynomial algorithm that computes the permanent of a nonnegative n x n matrix to within a multiplicative factor of e^n. To this end
An Optimal Algorithm for Monte Carlo Estimation
, 1995
"... A typical approach to estimate an unknown quantity is to design an experiment that produces a random variable Z distributed in [0; 1] with E[Z] = , run this experiment independently a number of times and use the average of the outcomes as the estimate. In this paper, we consider the case when no a ..."
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Cited by 53 (4 self)
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A typical approach to estimate an unknown quantity is to design an experiment that produces a random variable Z distributed in [0; 1] with E[Z] = , run this experiment independently a number of times and use the average of the outcomes as the estimate. In this paper, we consider the case when no a priori information about Z is known except that is distributed in [0; 1]. We describe an approximation algorithm AA which, given ffl and ffi, when running independent experiments with respect to any Z, produces an estimate that is within a factor 1 + ffl of with probability at least 1 \Gamma ffi. We prove that the expected number of experiments run by AA (which depends on Z) is optimal to within a constant factor for every Z. An announcement of these results appears in P. Dagum, D. Karp, M. Luby, S. Ross, "An optimal algorithm for MonteCarlo Estimation (extended abstract)", Proceedings of the Thirtysixth IEEE Symposium on Foundations of Computer Science, 1995, pp. 142149 [3]. Section ...
Markov Chains and Polynomial time Algorithms
, 1994
"... This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing ..."
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Cited by 48 (0 self)
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This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing the volumes of convex sets.
On Unapproximable Versions of NPComplete Problems
"... . We prove that all of Karp's 21 original NPcomplete problems have a version that's hard to approximate. These versions are obtained from the original problems by adding essentially the same, simple constraint. We further show that these problems are absurdly hard to approximate. In fact, no polyn ..."
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Cited by 35 (1 self)
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. We prove that all of Karp's 21 original NPcomplete problems have a version that's hard to approximate. These versions are obtained from the original problems by adding essentially the same, simple constraint. We further show that these problems are absurdly hard to approximate. In fact, no polynomialtime algorithm can even approximate log (k) of the magnitude of these problems to within any constant factor, where log (k) denotes the logarithm iterated k times, unless NP is recognized by slightly superpolynomial randomized machines. We use the same technique to improve the constant ffl such that MAX CLIQUE is hard to approximate to within a factor of n ffl . Finally, we show that it is even harder to approximate two counting problems: counting the number of satisfying assignments to a monotone 2SAT formula and computing the permanent of1,0,1 matrices. Key words. NPcomplete, unapproximable, randomized reduction, clique, counting problems, permanent, 2SAT AMS subject clas...
Polynomial Time Algorithms To Approximate Permanents And Mixed Discriminants Within A Simply Exponential Factor
 Random Structures & Algorithms
, 1999
"... We present real, complex, and quaternionic versions of a simple randomized polynomial time algorithm to approximate the permanent of a nonnegative matrix and, more generally, the mixed discriminant of positive semidefinite matrices. The algorithm provides an unbiased estimator, which, with high pro ..."
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Cited by 33 (3 self)
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We present real, complex, and quaternionic versions of a simple randomized polynomial time algorithm to approximate the permanent of a nonnegative matrix and, more generally, the mixed discriminant of positive semidefinite matrices. The algorithm provides an unbiased estimator, which, with high probability, approximates the true value within a factor of O(c n ), where n is the size of the matrix (matrices) and where c 0:28 for the real version, c 0:56 for the complex version and c 0:76 for the quaternionic version. We discuss possible extensions of our method as well as applications of mixed discriminants to problems of combinatorial counting.
Simple deterministic approximation algorithms for counting matchings
"... We construct a deterministic approximation algorithm for computing a permanent of a 0, 1 n by n matrix to within a multiplicative factor (1 + ǫ) n, for arbitrary ǫ> 0. When the graph underlying the matrix is a constant degree expander our algorithm runs in polynomial time (PTAS). In the general case ..."
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Cited by 22 (4 self)
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We construct a deterministic approximation algorithm for computing a permanent of a 0, 1 n by n matrix to within a multiplicative factor (1 + ǫ) n, for arbitrary ǫ> 0. When the graph underlying the matrix is a constant degree expander our algorithm runs in polynomial time (PTAS). In the general case the running time of the algorithm is exp(O(n 2 3 log 3 n)). For the class of graphs which are constant degree expanders the first result is an improvement over the best known approximation factor e n obtained in [LSW00]. Our results use a recently developed deterministic approximation algorithm for counting partial matchings of a graph [BGK +] and JerrumVazirani expander decomposition method of [JV96]. 1
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...