An entropy proof of Bregman's theorem

Abstract

Let A = (ai,j) be an n × n 0-1 matrix. Let S be the set of permutations σ of [n] such that ai,σ(i) = 1 for i = 1, 2, ..., n. Then, the permanent of A is perm(A) =def |S|. THEOREM (Brégman, Soviet Math. Dokl. 14 (1973), 945-949). If the number of 1's in row i of A is ri, then (Formula Presented) A short proof of this theorem was given by Schrijver (J. Combin. Theory Ser. A 25 (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. © 1997 Academic Press.