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...nt papers by Barvinok on counting zero/one matrices with prescribed row and column sums [22] and by Barvinok and Samorodnitsky on computing the partition function for perfect matchings in hypergraphs =-=[23]-=-. However, these papers pursue what would be called a mean-field theory approach in the physics literature [24]. An exception to the previous statement is Section 3.2 in [22], which contains Bethe-app...

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...octahedra in Ψ. We will now show how to replace φ+ φ′ by some ∆ ∈ ZXO(G) that is equivalent, in that ∂∆ = ∂(φ+ φ′), such that e does not appear in any octahedron in ∆. Let H be the r-complex on [r] × =-=[4]-=- generated by all r-sets of the form {(i, xi) : i ∈ [r]} for some x ∈ [4][r]. By identifying [4] with {0, 1}2 we can decompose H into 2r octahedra Ox, x ∈ {0, 1}r, where Ox is generated by all r-sets ...

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... article is prohibited.there have been only few results ([6], [7]) on approximate counting in hypergraphs before. The only other paper devoted to counting matchings in hypergraphs we are aware of is =-=[1]-=-, where Barvinok and Samorodnitsky compute the partition function for matchings in hypergraphs under some restrictions on the weights of edges. In particular they are able to distinguish in polynomial...

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...lynomial time algorithm based on scaling approximates the permanent of a given n× n real matrix A = (aij) satisfying ≤ aij ≤ 1 for all i, j within a multiplicative factor of nκ() for some κ() > 0 =-=[BS11]-=-. (1.5) Ramifications. In Section 4, we discuss how our approach can be used for computing hafnians of symmetric matrices and multidimensional permanents of tensors. The same approach can be used for ...

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...terested in computing permanents of a rather restricted class of matrices. Namely, let us fix a δ > 0 and suppose that (1.1.1) δ ≤ aij ≤ 1 for all i, j. Then the scaling algorithm of [L+00], see also =-=[BS11]-=-, approximates per A within a factor of n O(1) , where the implied constant in the O(1) notation depends on δ. 2The advantage of the algorithm of [L+00] is that beside being polynomial time, it is de...

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...scaling algorithm [14] transforms the adjacency matrix of a graph to an almost doubly stochastic one. Yet, a non-trivial lower estimate of the hafnian of a doubly stochastic matrix is impossible, see =-=[3]-=-. Also, in contrast with the computation of the permanent, [12] points out that the proof of convergence of the MCMC algorithm breaks down for the approximate computation of the hafnian (unless the mi...

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...ew line of research, as there have been only few results ([5], [6]) on approximate counting in hypergraphs before. The only other paper devoted to counting matchings in hypergraphs we are aware of is =-=[1]-=-, where Barvinok and Samorodnitsky compute the partition function for matchings in hypergraphs under some restrictions on the weights of edges. In particular they are able to distinguish in polynomial...

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...lynomial time algorithm based on scaling approximates the permanent of a given n× n real matrix A = (aij) satisfying ǫ ≤ aij ≤ 1 for all i, j within a multiplicative factor of nκ(ǫ) for some κ(ǫ) > 0 =-=[BS11]-=-. (1.5) Ramifications. In Section 4, we discuss how our approach can be used for computing hafnians of symmetric matrices and multidimensional permanents of tensors. The same approach can be used for ...