An Entropy Proof of Bregmanʹs Theorem

LetA=(ai, j) be ann×n0-1 matrix. LetSbe the set of permutationsσof [n] such thatai, σ(i)=1 fori=1, 2, …, n. Then, the permanent ofAis perm(A)=def|S|. Theorem(Brégman,Soviet Math. Dokl.14(1973), 945–949). If the number of 1ʹs in rowiofAisri, then[formula]A short proof of this theorem was given by Schrijver (J. Combin. Theory Ser. A25(1978), 80–83); Alon and Spencer (“The Probabilistic Method,” Wiley/Interscience, New York, 1992), obtained a similar proof by analyzing a randomized procedure for estimating the permanent. In this note we present a proof based on entropy.