On the Birkhoff-von Neumann decomposition

Michele Benzi, Alex Pothen and I have been making use of the celebrated Birkhoff-von Neumann theorem on doubly stochastic matrices. The theorem says that any doubly stochastic matrix can be written as a convex combination of a finitely many permutation matrices. Formally, let \mathbf{A} be a doubly stochastic matrix. Then,

\mathbf{A}=\sum_{j=1}^k \alpha_j \mathbf{P}_j\;,

where \alpha_j>0, \sum_j \alpha_j=1 and each \mathbf{P}_j is a permutation matrix.

Given this formulation, one wonders if the decomposition is unique. Well, the answer is “No”. Then, one asks what can be said about the number k. And this is the main topic of this post.

Richard A. Brualdi [1] discusses many things among which a lower bound and an upper bound on k. The minimum number of permutation matrices is equal to the maximum cardinality of a set of nonzeros positions of \mathbf{A} no two of which could appear together in a single permutation matrix in the pattern of \mathbf{A}. An easy lower bound is then equal to the maximum number of nonzeros in a row or a column of \mathbf{A}. The upper bound is \mathrm{nnz}(\mathbf{A})-2n+2, for a fully indecomposable matrix \mathbf{A} with \mathrm{nnz}(\mathbf{A}) nonzeros; more generally if there are b fully indecomposable blocks, then the upper bound is \mathrm{nnz}(\mathbf{A})-2n+b+1.

What about the minimum number of permutation matrices? It turns out that this is an NP-complete problem. Let’s state it in the form of a standard NP-completeness result.

Input:  A doubly stochastic matrix \mathbf{A}.
Output: A Birkhoff-von Neumann decomposition of \mathbf{A} as \mathbf{A} = \alpha_1\mathbf{P}_1 + \alpha_2\mathbf{P}_2 + \cdots +\alpha_k\mathbf{P}_k.
Measure: The number k of permutation matrices in the decomposition.

The NP-completeness of the decision version of this problem is shown in a recent technical report [2].


  1. Richard A. Brualdi, Notes on the Birkhoff algorithm for doubly stochastic matrices, Canadian Mathematical Bulletin, 25(2), 191–199, 1982 (doi).
  2. Fanny Dufossé and Bora Uçar, Notes on Birkhoff-von Neumann decomposition of doubly stochastic matrices, Technical Report, RR-8852, Inria Grenoble Rhône-Alpes, 2016 (link).

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