3. Permanents

Introduced in 1812 by Binet and Cauchy, permanents are of interest to combinatorics, as they enumerate perfect matchings in bipartite graphs, to physics as they compute certain integrals and to computer science as they occupy a special place in the computational complexity hierarchy. This is our first example of a partition function and we demonstrate in detail how various approaches work. Connections with \({\mathbb {H}}\)-stable polynomials lead, in particular, to an elegant proof of the van der Waerden lower bound for the permanent of a doubly stochastic matrix. Combining it with the Bregman - Minc upper bound, we show that permanents of doubly stochastic matrices are strongly concentrated. Via matrix scaling, this leads to an efficient approximation of the permanent of non-negative matrices by a function with many convenient properties: it is easily computable, log-concave and generally amenable to analysis. As an application of the interpolation method, we show how to approximate permanents of a reasonably wide class of complex matrices and also obtain approximations of logarithms of permanents of positive matrices by low degree polynomials.