Notes on the Birkhoff Algorithm for Doubly Stochastic Matrices

Abstract

The purpose of this note is to tie together some results concerning doubly stochastic matrices and their representations as convex combinations of permutation matrices.

Further notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices

“…Marcus-Ree Theorem [1] states that for a dense matrix, there are decompositions with k ≤ n 2 −2n+2 permutation matrices. For a sparse, fully indecomposable matrix with τ nonzeros, the same result holds [2,3] with k ≤ τ −2n+2. Brualdi [3] gives lower bounds on the number of permutation matrices in any BvN decomposition of a given matrix.…”

“…For a sparse, fully indecomposable matrix with τ nonzeros, the same result holds [2,3] with k ≤ τ −2n+2. Brualdi [3] gives lower bounds on the number of permutation matrices in any BvN decomposition of a given matrix. A recent work [4] shows that the problem of finding a BvN decomposition with the smallest number k of permutation matrices is strongly NP-complete.…”

“…In other words, we can write A = P (1) + P (2) + (n − 2) · P (3) , and see that A has a BvN decomposition with three permutation matrices. We note that each row and column of A contains three nonzeros; and hence three is the smallest number of permutation matrices in a BvN decomposition of A.…”

An ordinal evaluation of categorical judgement data by random utilities and a corresponding correlation analysis

“…One can think of several possibilities. One was suggested by Brualdi (1982) in connection with the polytope of doubly stochastic matrices. He proposed to use the representation which maximizes entropy.…”

Comments on a Paper of R. A. Brualdi

“…A representation of a doubly stochastic matrix D as a convex combination of permutation matrices P i9 obtained by applying the Birkhoff algorithm, is called a Birkhoff representation of D (see [1] for the details of the Birkhoff algorithm).…”