Brégman ’ s theorem and extensions

Minc conjectured, and Brégman proved, a sharp upper bound on the permanent of an n by n 0-1 matrix with given row sums (equivalently, on the number of perfect matchings in a bipartite graph with each partition class having size n and with fixed degree sequence for one of the two classes). Here we present Radhakrishnan’s entropy proof of Brégman’s theorem, and Alon and Friedland’s proof of an analogous statement for graphs that are not necessarily bipartite. We also discuss progress towards the Upper Matching conjecture of Friedland, Krop and Markström, which extends Brégman’s theorem to arbitrary matchings.