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13
The Bethe Permanent of a NonNegative Matrix
, 2010
"... It has recently been observed that the permanent of a nonnegative matrix, i.e., of a matrix containing only nonnegative real entries, can very well be approximated by solving a certain Bethe free energy minimization problem with the help of the sumproduct algorithm. We call the resulting approxima ..."
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It has recently been observed that the permanent of a nonnegative matrix, i.e., of a matrix containing only nonnegative real entries, can very well be approximated by solving a certain Bethe free energy minimization problem with the help of the sumproduct algorithm. We call the resulting approximation of the permanent the Bethe permanent. In this paper we give reasons why this approach to approximating the permanent works well. Namely, we show that the Bethe free energy is a convex function and that the sumproduct algorithm finds its minimum efficiently. We also show that the permanent is lower bounded by the Bethe permanent, and we list some empirical evidence that the permanent is upper bounded by some constant (that modestly grows with the matrix size) times the Bethe permanent. Part of these results are obtained by a combinatorial characterization of the Bethe permanent in terms of permanents of socalled lifted versions of the matrix under consideration. We conclude the paper with some conjectures about permanentbased pseudocodewords and permanentbased kernels, and we comment on possibilities to modify the Bethe permanent so that it approximates the permanent even better.
The existence of designs
"... Abstract. We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition. 1. ..."
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Abstract. We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition. 1.
Approximate Counting of Matchings in Sparse Uniform Hypergraphs
"... In this paper we give a fully polynomial randomized approximation scheme (FPRAS) for the number of matchings in kuniform hypergraphs whose intersection graphs contain few claws. Our method gives a generalization of the canonical path method of Jerrum and Sinclair to hypergraphs satisfying a local r ..."
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In this paper we give a fully polynomial randomized approximation scheme (FPRAS) for the number of matchings in kuniform hypergraphs whose intersection graphs contain few claws. Our method gives a generalization of the canonical path method of Jerrum and Sinclair to hypergraphs satisfying a local restriction. The proof depends on an application of the Euler tour technique for the canonical paths of the underlying Markov chains. On the other hand, we prove that it is NPhard to approximate the number of matchings even for the class of 2regular, linear, kuniform hypergraphs, for all k ≥ 6, without the above restriction. 1
COMPUTING THE PERMANENT OF (SOME) COMPLEX MATRICES
, 2014
"... Abstract. We present a deterministic algorithm, which, for any given 0 < < 1 and an n × n real or complex matrix A = (aij) such that aij − 1  ≤ 0.19 for all i, j computes the permanent of A within relative error in nO(lnn−ln ) time. The method can be extended to computing hafnians and mul ..."
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Abstract. We present a deterministic algorithm, which, for any given 0 < < 1 and an n × n real or complex matrix A = (aij) such that aij − 1  ≤ 0.19 for all i, j computes the permanent of A within relative error in nO(lnn−ln ) time. The method can be extended to computing hafnians and multidimensional permanents. 1. Introduction and
PARTITION FUNCTIONS FOR DENSE INSTANCES OF COMBINATORIAL ENUMERATION PROBLEMS
, 2013
"... Abstract. Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds: cycle covers, closed walks, spanning trees. We show th ..."
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Abstract. Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds: cycle covers, closed walks, spanning trees. We show that if the weights of the edges of the graph are within a constant factor, fixed in advance, of each other then the bulk of the partition function is concentrated on the subsets of a particularly simple structure: cycle covers with few cycles, walks that visit every vertex only few times, and spanning trees with small degree of every vertex. This allows us to construct a polynomial time algorithm to separate graphs with many Hamiltonian cycles from graphs that are sufficiently far from Hamiltonian. 1. Introduction and
Approximate Counting of Matchings in Sparse Hypergraphs
"... In this paper we give a fully polynomial randomized approximation scheme (FPRAS) for the number of all matchings in hypergraphs belonging to a class of sparse, uniform hypergraphs. Our method is based on a generalization of the canonical path method to the case of uniform hypergraphs. ..."
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In this paper we give a fully polynomial randomized approximation scheme (FPRAS) for the number of all matchings in hypergraphs belonging to a class of sparse, uniform hypergraphs. Our method is based on a generalization of the canonical path method to the case of uniform hypergraphs.
RANDOM GAUSSIAN MATRICES AND HAFNIAN ESTIMATORS
, 2014
"... We analyze the behavior of the Barvinok estimator of the hafnian of even dimension, symmetric matrices with non negative entries. We introduce a condition under which the Barvinok estimator achieves subexponential errors, and show that this condition is almost optimal. Using that hafnians count ..."
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We analyze the behavior of the Barvinok estimator of the hafnian of even dimension, symmetric matrices with non negative entries. We introduce a condition under which the Barvinok estimator achieves subexponential errors, and show that this condition is almost optimal. Using that hafnians count the number of perfect matchings in graphs, we conclude that Barvinok’s estimator gives a polynomialtime algorithm for the approximate (up to subexponential errors) evaluation of the number of perfect matchings.