## A Polynomial-Time Approximation Algorithm for the Permanent of a Matrix with Non-Negative Entries (2004)

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Venue: | Journal of the ACM |

Citations: | 314 - 23 self |

### BibTeX

@ARTICLE{Jerrum04apolynomial-time,

author = {Mark Jerrum and Alistair Sinclair and Eric Vigoda},

title = {A Polynomial-Time Approximation Algorithm for the Permanent of a Matrix with Non-Negative Entries},

journal = {Journal of the ACM},

year = {2004},

pages = {671--697}

}

### Years of Citing Articles

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### Abstract

Abstract. We present a polynomial-time randomized algorithm for estimating the permanent of an arbitrary n ×n matrix with nonnegative entries. This algorithm—technically a “fully-polynomial 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

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Citation Context ...ve been observed by several authors: see, e.g., [8, 1, 23, 25]. (ii) Theorem 6 has a converse, which says that if a Markov chain is rapidly mixing then its conductance cannot be too small: see, e.g., =-=[23, 27, 28]-=-. Thus the conductance provides a characterisation of the rapid mixing property. Theorem 6 allows us to investigate the rate of convergence of a reversible chain by examining its transition structure,... |

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Citation Context ...ently intractable in exact form, but for which an efficient approximation algorithm exists. This is an intriguing class of problems, and includes the problems of computing the volume of a convex body =-=[9]-=-, the partition function of a monomer-dimer system [16] and the permanent of a large class of 0--1 matrices [16]. Our algorithm is also of interest in its own right as a further application of the gen... |

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Citation Context ...art of the above proof is the eigenvalue bound (23). This is a discrete analogue of Cheeger's inequality for Riemannian manifolds [5]. Related bounds have been observed by several authors: see, e.g., =-=[8, 1, 23, 25]-=-. (ii) Theorem 6 has a converse, which says that if a Markov chain is rapidly mixing then its conductance cannot be too small: see, e.g., [23, 27, 28]. Thus the conductance provides a characterisation... |

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Citation Context ...ient computational solutions to these problems has proved extremely hard and has generated a vast body of literature. A major breakthrough was achieved in the early 1960s by Kasteleyn [19] and Fisher =-=[11]-=-, who reduced the problem of computing Z for any planar Ising system (i.e., one whose graph ([n]; E) of non-zero interactions is planar) to the evaluation of a certain determinant. This must rank as o... |

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Citation Context ... mathematics communities since it was first introduced by Lenz [24] and Ising [14] in the early 1920s. We will not present a detailed historical account here: a very readable survey is given by Cipra =-=[6]-=-, while Welsh [30] sets the Ising model in the context of other combinatorial problems in statistical physics. The problem is easily stated. Consider a collection of sites [n] = f0; 1; : : : ; n \Gamm... |

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Citation Context ... other words, MaxCut 2 BPP. From this it would follow---since MaxCut is NP-complete and BPP is closed under polynomial time reductions---that NP ` BPP. However, the inclusion NP ` BPP entails RP = NP =-=[22]-=-. Our final theorem states that Ising is a complete problem for the class #P. Thus a polynomial time algorithm which solved it exactly would yield similar algorithms for a range of presumably intracta... |

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Citation Context ...ivial requirement, since the number of states is exponentially large. Recent developments have provided appropriate analytical tools for establishing the rapid mixing property for chains of this kind =-=[27, 29, 7, 28]-=-. The Markov chain simulation approach to the Ising model is far from new: under the name of the Monte Carlo method, this technique has been applied extensively to a whole range of problems in statist... |

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Citation Context ...umber of cut-sets in G of maximum size. The following is a slight extension of a known result. Lemma 13 MaxCut is NP-complete, and #MaxCut is #P-complete. Proof NP-completeness of MaxCut is proved in =-=[13]. The redu-=-ctions used there are not "parsimonious" [12, p. 169], and hence do not immediately imply #P-completeness of #MaxCut. As usual, however, the reductions (given in the proofs of Theorems 1.1 a... |

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Citation Context ...l problem of statistical physics. Generally known as the Ising model, the problem has been the focus of much attention in the physics and mathematics communities since it was first introduced by Lenz =-=[24]-=- and Ising [14] in the early 1920s. We will not present a detailed historical account here: a very readable survey is given by Cipra [6], while Welsh [30] sets the Ising model in the context of other ... |

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Citation Context ...with the appropriate equilibrium distribution which is rapidly mixing. Thus the Markov chain approach can be made to work efficiently in the new domain. The above transformation is a classical result =-=[26], often kn-=-own as the "hightemperature expansion" of the Ising model partition function. However, the idea of viewing the graphs in this expansion as a statistical mechanical system which forms the bas... |