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## 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: | 426 - 27 self |

### Citations

14082 | Computers and Intractability: A Guide to the Theory of NP-Completeness - Garey, Johnson - 1979 |

3282 | An introduction to probability theory and its applications. Vol II. - Feller - 1966 |

2196 | Randomized algorithms - Motwani, Raghavan - 1995 |

1972 | The Probabilistic Method - Alon, Spencer - 1992 |

651 | The complexity of computing the permanent - Valiant - 1979 |

426 | The Art of Computer Programming, Volume 1 (Fundamental Algorithms), 2nd Edition - Knuth - 1973 |

400 | Eigenvalues and expanders.
- Alon
- 1986
(Show Context)
Citation Context ...lower bound on the expectation E �� 0 (f). Such a bound is provided by the next lemma, whose proof we defer to the end of the section. Lemma 4 Let �� 0 and �� 1 be arbitrary real numbers i=-=n the range [0; 1] satisfying �� 1-=- ! �� 0s�� 1 + n \Gamma1 . Then the ratio Z 0 (�� 1 )=Z 0 (�� 0 ) is bounded below by 1=10. Lemma 4 suggests that we should be able to bootstrap the known value Z 0 (1) to the desired ... |

348 | Geometric bounds for eigenvalues of Markov chains,
- Diaconis, Stroock
- 1991
(Show Context)
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... |

345 | Approximating the permanent.
- Jerrum, Sinclair
- 1989
(Show Context)
Citation Context ...cient 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 general technique of simulating an ergodic stochastic proc... |

341 | Random generation of combinatorial structures from a uniform distribution. - Jerrum, Valiant, et al. - 1986 |

336 |
A lower bound for the smallest eigenvalue of the Laplacian, in:
- Cheeger
- 1969
(Show Context)
Citation Context ...ing (22) and (23) together establishes the theorem. Remarks (i) The heart 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... |

317 | Approximate counting, uniform generation and rapidly mixing markov chains.
- Sinclair, Jerrum
- 1989
(Show Context)
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... |

310 |
Beitrag zur Theorie des Ferromagnetismus,” Zeitschrift fur
- Ising
- 1925
(Show Context)
Citation Context ...atistical 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 combinatorial p... |

276 | The Markov chain Monte Carlo method: an approach to approximate counting and integration - Jerrum, Sinclair - 1996 |

274 |
Optimization by Simulated Annealing. Science 220
- Kirkpatrick, Gelatt, et al.
- 1983
(Show Context)
Citation Context ... which is often used in practice, is to allow transitions to occur between spin configurations which differ in just one component, and choose transition probabilities according to the Metropolis rule =-=[20]-=-. If the resulting Markov chain is rapidly mixing, that is, if it converges rapidly to the stationary distribution regardless of the choice of initial state, then it can be used effectively to sample ... |

205 | Improved bounds for mixing rates of Markov chains and multicommodity flow.
- Sinclair
- 1992
(Show Context)
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,... |

204 | Comparison theorems for reversible markov chains. - Diaconis, Saloff-Coste - 1993 |

192 | The statistics of dimers on a lattice I: The number of dimer arrangements on a quadratic lattice, - Kasteleyn - 1961 |

178 | On the computational complexity of ising spin glass models. - Barahona - 1982 |

166 | A random polynomial time algorithm for approximating the volume of convex bodies.
- Dyer, Frieze, et al.
- 1991
(Show Context)
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... |

163 |
Di erence equations, isoperimetric inequality and transience of certain random walks,
- Dodziuk
- 1984
(Show Context)
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... |

144 |
Monte-carlo algorithms for enumeration and reliability problems.
- Karp, Luby
- 1983
(Show Context)
Citation Context ...we explain how to use samples produced by this procedure to obtain a reliable approximation of Z . Our definition of efficient approximation algorithm is a very demanding one, following Karp and Luby =-=[18]-=- and others. For non-negative real numbers a, ~ a, ffl , we say that ~ a approximates a within ratio 1 + ffl if a(1 + ffl) \Gamma1s~asa(1 + ffl) . Let f be any function from problem instances to real ... |

131 |
Dimer statistics and phase transitions.
- Kasteleyn
- 1963
(Show Context)
Citation Context ...search for efficient 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. Thi... |

104 | Counting, Sampling and Integrating: Algorithms and Complexity (Lectures in Mathematics. - Jerrum - 2003 |

103 |
Isoperimetric number of graphs,
- Mohar
- 1989
(Show Context)
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... |

98 | The complexity of computing the permanent, Theoretical computer science 8 - Valiant - 1979 |

97 | A Chernoff bound for random walks on expander graphs - Gillman - 1998 |

80 | On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing, - Aldous - 1987 |

75 | A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents, - Linial, Samorodnitsky, et al. - 2000 |

73 |
On the dimer solution of planar ising models.
- Fisher
- 1966
(Show Context)
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... |

72 | How hard is it to marry at random? (on the approximation of the permanent - Broder - 1986 |

53 | Two-dimensional monomer-dimer systems are computationally intractable. - Jerrum - 1987 |

51 |
An Introduction to the Ising Model.
- Cipra
- 1987
(Show Context)
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... |

49 | A short proof of the factor theorem for finite graphs, - Tutte - 1954 |

48 | Chernoff-type bound for finite Markov chains - Lezaud - 1998 |

39 | Polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor, - Barvinok - 1999 |

34 |
Beitrag zum Verstandnis der magnetishen Erscheinungen in festen Korpern.
- Lenz
- 1920
(Show Context)
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 ... |

34 | Fast uniform generation of regular graphs. - Jerrum, Sinclair - 1990 |

30 | Vazirani: Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science 43 - Jerrum, Valiant, et al. - 1986 |

29 |
Some observations on the probabilistic algorithms and NP-hard problems
- Ko
- 1982
(Show Context)
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... |

29 | A mildly exponential approximation algorithm for the permanent. Algorithmica 16:392–401, - Jerrum, Vazirani - 1996 |

25 |
Bounds on the L spectrum for Markov chains and Markov processes: A generalization of Cheeger’s inequality
- LAWLER, SOKAL
- 1988
(Show Context)
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... |

24 | The computational complexity of some classical problems from statistical physics, Disorder in Physical Systems, - Welsh - 1990 |

20 |
Randomised algorithms for counting and generating combinatorial structures,
- Sinclair
- 1993
(Show Context)
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... |

19 | On the number of Eulerian orientations of a graph,” - Mihail, Winkler - 1996 |

17 |
Some NP-complete problems
- Garey, Graham, et al.
- 1976
(Show Context)
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... |

16 | On the number of Eulerian orientations of a graph,” Algorithmica 16 - Mihail, Winkler - 1996 |

13 | Fast uniform generation of regular graphs, Theoretical Computer Science 73 - Jerrum, Sinclair - 1990 |

10 |
On the theory of the Ising model of ferromagnetism,
- Newell, Montroll
- 1953
(Show Context)
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... |

9 | An O(n2) bound for the relaxation time of a Markov chain on cladograms. Random Structures and Algorithms - Schweinsberg - 2000 |

8 | Avi Wigderson. A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents - Linial, Samorodnitsky |

6 |
Monte-Carlo investigations of phase transitions and critical phenomena”, in Phase Transitions and
- Binder
- 1976
(Show Context)
Citation Context ...tion 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 statistical physics (see, e.g., =-=[4]-=-). The problem with the approach, however, is that it appears very difficult to define a Markov chain on Ising spin configurations oe which is rapidly mixing; indeed, the chains which are frequently u... |

5 | Approximating the number of dimer coverings of a lattice - Kenyon, Randall, et al. - 1996 |

3 | Jerrum and Umesh Vazirani, A mildly exponential approximation algorithm for the permanent, Algorithmica 16 - Mark - 1996 |

1 |
Random generation of combinatorial structues from a uniform distribution, Theoretical Computer Science 43
- Jerrum, Valiant, et al.
- 1986
(Show Context)
Citation Context ...h probability at least 3=4. This is because a failure probability of 1=4 can be reduced to any desired value ffi ? 0 by performing only O(log ffi \Gamma1 ) trials and taking the median of the results =-=[17]-=-. (This claim is also justified in the proof of Lemma 3 below.) Of course, it is not enough just to obtain an accurate result with high reliability; the result must also be obtained efficiently. Accor... |

1 | A Polynomial-Time Approximation Algorithm 697 - MINC - 1982 |