M. Jerrum and A. Sinclair. Approximating the permanent. SIAM Journal of Computing, 18:1149--1178, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....along the lines of McCaskill [18] should be possible, but would be computationally expensive and limited by the range of structures handled by the Rivas and Eddy algorithm. It may be possible to approximate the partition function via the Markov chain monte carlo method of Jerrum and Sinclair [14]. Finally, we note that secondary structures can also form between two or more RNA or DNA molecules in solution, so a natural generalization of the problem discussed so far is to predict the mfe secondary structure formed by two or more input molecules. Conceptually, the thermodynamic model for a ....

M. Jerrum and A. Sinclair, "Approximating the permanent", SIAM Journal on Computing 18, 1989, 1149--1178.

....if every vertex of G is incident to precisely one edge from Y . Let F ae 2 be the set of all perfect matchings in G. The problem of computing or estimating jF j efficiently is one of the hardest and most intriguing problems of combinatorial counting, see, for example, Lov asz and Plummer 86] Jerrum and Sinclair 89] Jerrum 95] Jerrum and Sinclair 97] and [Jerrum, et al. 00] It is known that the problem of exact counting of perfect matchings is hard. It belongs to the class of # P hard problems, see Chapter 18 of [Papadimitriou 94] for discussion of computational complexity in enumeration problems. ....

.... this, a Markov chain on the set F is generated, so that it converges rapidly to the uniform distribution on F (see [Jerrum and Sinclair 97] for a survey) Spectacular successes of this approach are finding a polynomial time randomized algorithm to count matchings of all sizes in a given graph [Jerrum and Sinclair 89] and to count perfect matchings in a given bipartite graph [Jerrum et al. 00] both within any prescribed relative error. When the Markov chain approach works, it produces incomparably better results than the method of this paper. However, for many important counting problems, some of which are ....

M. Jerrum and A. Sinclair, Approximating the permanent, SIAM J. Comput., 18 (1989), no. 6, 1149--1178.

....in planar graphs [7] uses O (n s) arithmetic operations, the fastest known general algorithm for the exact computation of the permanent needs O(n2 n) operations ( 6, 12] However, the determinant of an n x n matrix can be evaluated in O(n 3) arithmetic operations using Gaussian elimination. In [4], only the permanent of the matrix A with 0 1 entries aij is of interest instead that of the general matrix. The calculation of per(A) is equivalent to counting the number of perfect matchings in the bipartite graph G = U,V,E) in which U = V = n] and (i,j) 6 E if and only aij = 1. The ....

....is rapidly m9cing. In [2] Broder try to use coupling [1] to prove that the perfect matching chains are rapidly mixing when the minimum vertex degree of the bipartite graph is at least n 2. However, an uncorrectable error was found by Mihail[9] such that Broder withdraw his proof. This paper [4] is the second important step, in which Jerrum and Sinclair prove the Markov chain used in [2] to be rapidly mixing. Their method depends on the conductance of the underlying graph. A finite ergodic Markov chain is rapidly mixing if and only if the conductance of the graph is sufficiently big. ....

M. Jerrum and A. Sinclair. Approximating the permanent. SIAM J. Comput., 18(6):1149-1178, 1989.

....Markov chain will generate a graph with the given degree sequence uniformly at random. WewouldbeinterestedinaMarkovchainwhich is arbitrarily close to the uniform distribution after simulating a polynomial number of steps (see [35] for details) Similar questions have been considered elsewhere [35, 22, 21, 23] without the connectivity requirement. In particular, it is known that uniform generation of a simple graph with a given degree sequence d = d 1 . dn reduces to uniform generation of a perfect matching of the following graph Md [28] Fo r each 1 n, Md contains a complete bipartite ....

....graph. This does not include the case of arbitrary power law graphs, and hence that theory does not apply. Indeed, generating a random graph that meets an arbitrary degree sequence d is a major open problem, at least since the original paper of Jerrum and Sinclair on approximating permanents [22]. In addition, we note here that the problem of rapid mixing of connected realizations is strictly harder than that of arbitrary realizations without the connectivity requirement, as indicated by the following reduction: For a degree sequence d = d 1 . dn , introduce an additional ....

M. Jerrum and A. Sinclair. Approximating the permanent. SIAM J. of Computing, 18:1149--1178, 1989.

....adjacency matrix of a random bipartite multigraph and then, from the standard theory of spectra of graphs[5] we have that i = i 0 with probability 1 as 0 and jC i j 1. Below we give a formal justification of this by showing that a quantity that captures this property, the conductance [19] (equivalently, expansion) of B i B i is high. The conductance of an undirected edge weighted graph G = V; E) is min SaeV i2S;j2S wt(i; j) minfjSj; jSjg Let x ; x ; x be random documents picked from the topic T i . Then we will show that the conductance is Omega Gamma ....

....corpus model. Suppose that documents are nodes in a graph, and weights on the edges capture conceptual proximity of two documents (for example, this distance matrix could be derived from, or in fact coincide with, AA ) Then a topic is defined implicitly as a subgraph with high conductance [19], a concept of connectivity which seems very appropriate in this context. We can prove that, under an assumption similar to ffl separability, spectral analysis of the graph can identify the topics in this model as well (proof omitted from this abstract) Theorem 5 If the corpus consists of k ....

M. Jerrum and A. Sinclair. Approximating the permanent. Siam J. Comp. 18, pp. 1149-1178, (1989).

.... simulation has emerged as a powerful algorithmic tool and has had a profound impact on random sampling and approximate counting [18] Among its numerous applications are estimating the volume of convex bodies [10] see also [23] for recent progress on this problem) and approximating the permanent [17]. Very recently, Jerrum, Sinclair and Vigoda [19] used this approach to solve the long standing open problem of approximating the permanent for nonnegative matrices. Despite the fact that our quantum walks are easy to describe, certain variants (in particular the case where the walk has absorbing ....

M. Jetrum and A. Sinclair. Approximating the permanent. SIAM Journal on Computing, 18(6):1149-1178, December 1989.

....mixes rapidly if the graph is dense enough (e.g. if each row and column of A contains at least n 2 ones) This yields a polynomial time randomized approximation algorithm for computing the number of perfect matchings of dense bipartite graphs. For details, the reader is invited to consult [J S] or [Sin] 6 Infinite graphs In this section we show how to extend the results of previous sections to infinite locally finite graphs. Most of the results can be derived for weighted graphs where the local finiteness condition is replaced by the requirement that d v for each v V (G) ....

M. Jerrum, A. Sinclair, Approximating the permanent, SIAM J. Comput. 18 (1989) 1149-- 1178.

.... Vega and Kenyon [FdVKe98] It is also a very interesting artifact that the recent successes in design of the polynomial time approximation schemes for dense optimization problems parallel the successes of the past attacks on dense approximate counting problems, Broder [B86] Jerrum and Sinclair [JS89], Dyer, Frieze, Jerrum [DFJ94] and Alon, Frieze and Welsh [AFW95] 2 MAX SNP and Dense MAX SNP Classes, and BEYOND We consider in this Section the dense instances of the MAX SNP class of optimization problems introduced by Papadimitriou and Yannakakis [PY91] MAXSNP class contains ....

M. R. Jerrum and A. Sinclair, Approximating the Permanent, SIAM J. Comput. 18 (1989), pp. 1149-1178.

.... A function f : Sigma N is said to have a fully polynomial time randomized approximation scheme (fpras) 14, 13] if there is a PPTM M such that for all x 2 Sigma and c 0, taking ffl : 1=c: Pr u f(x) 1 ffl) M(hx; 0 c i; u) f(x) 1 ffl) # 3=4: 2) Jerrum and Sinclair [12] showed that the permanent function for dense 0 1 matrices, which is still #P complete, has an fpras. Note that M is multi valued. We observe that the approximation can be done by a total function which is at most 2 valued. The 3=4 here and in (2) can be amplified to give exponentially ....

M. Jerrum and A. Sinclair. Approximating the permanent. SIAM J. Comput., 18:1149--1178, 1989.

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M. Jerrum and A. Sinclair, "Approximating the Permanent," SIAM Journal on Computing 18 (1989), pp. 1149--1178.

....that a given chain achieves the right stationary distribution [20, 24] Aldous [1] was the first to use coupling to show rapid mixing. We will describe this technique in detail shortly. However, as pointed out by Mihail [23] there was an error in Broder s coupling argument. Jerrum and Sinclair [17] later showed that Broder s chain (actually a slight variant of it) mixes rapidly using a completely different technique, which involved showing that the underlying graph had large conductance, by demonstrating the existence of several canonical paths with low edge congestion. The Coupling ....

....possible to show rapid mixing (if the chain in question indeed mixes rapidly) by setting up an appropriate non causal coupling and bounding its coupling time. However, the latter task is typically hard. In this paper, we study only causal couplings. Coupling v s Conductance. Jerrum and Sinclair [17] showed that a Markov chain mixes rapidly if and only if it has large conductance. Our original aim was to study the following question: is the coupling method as powerful as the conductance method for showing rapid mixing In other words, if the Markov chain in question has large conductance, ....

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M. Jerrum and A. Sinclair, Approximating the permanent, SIAM Journal on Computing 18, 1989, pp. 1149-1178.

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M. Jerrum and A. Sinclair. Approximating the permanent. SIAM Journal of Computing, 18:1149--1178, 1989.

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Mark Jerrum and Alistair Sinclair. Approximating the Permanent. SIAM J. Computing 18(1989), pp. 1149-1178.

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M. Jerrum and A. Sinclair, Approximating the permanent, SIAM J. on Computing, 18:1149-1178, 1989.

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M. Jerrum and A. Sinclair, Approximating the permanent, SIAM J. on Computing, 18:1149-1178, 1989. 6

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M. Jerrum and A. Sinclair. Approximating the permanent. SIAM J. Comput., 18(6):1149-1178, 1989.

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M. Jerrum and A. Sinclair. Approximating the permanent. SIAM J. Comput., 18(6):1149-1178, 1989.

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M. Jerrum and A. Sinclair, Approximating the permanent, SIAM J. Comput., 18, 1149-1178, 1989.

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M. Jerrum and A. Sinclair. Approximating the permanent. SIAM Journal on Computing, 18(6):1149-- 1178, 1989.

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Mark Jerrum and Alistair Sinclair. Approximating the Permanent. SIAM Journal of Computing, 18:1149--1178, 1989.

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M. Jerrum and A. Sinclair. Approximating the permanent. SIAM J. on Computing, 18:1149-1178, 1989.

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M. Jerrum and A. Sinclair, Approximating the permanent, SIAM J. Comput. 18 (1988) 1149-1178.

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Jerrum, M. and Sinclair, A. (1989) Approximating the permanent. SIAM Journal on Computing 18(6) 1149-1178 (December).

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M. Jerrum and A. Sinclair, Approximating the permanent, SIAM J. Comput. 18 (1988) 1149-1178.

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M. Jerrum, A. Sinclair, "Approximating the Permanent ", SIAM J. Computing, 18(6):1149 1178, December 1989.

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