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It has recently been observed that the permanent of a non-negative matrix, i.e., of a matrix containing only non-negative real entries, can very well be approximated by solving a certain Bethe free energy minimization problem with the help of the sum-product 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 sum-product 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 so-called lifted versions of the matrix under consideration. We conclude the paper with some conjectures about permanent-based pseudo-codewords and permanent-based kernels, and we comment on possibilities to modify the Bethe permanent so that it approximates the permanent even better.

The Bethe permanent of a non-negative matrix