Bregman's Theorem

In discrete mathematics, the Bregman–Minc inequality, or Bregman's theorem, allows one to estimate the permanent of a binary matrix via its row or column sums. The inequality was conjectured in 1963 by Henryk Minc and first proved in 1973 by Lev M. Bregman. Further entropy-based proofs have been given by Alexander Schrijver and Jaikumar Radhakrishnan. The Bregman–Minc inequality is used, for example, in graph theory to obtain upper bounds for the number of perfect matchings in a bipartite graph.

The permanent of a square binary matrix ${\displaystyle A=(a_{ij})}$ of size ${\displaystyle n}$ with row sums ${\displaystyle r_{i}=a_{i1}+\cdots +a_{in}}$ for ${\displaystyle i=1,\ldots ,n}$ can be estimated by

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