Minimum Birkhoff-von Neumann Decomposition

Motivated by the applications in routing in data centers, we study the problem of expressing an n x n doubly stochastic matrix as a linear combination using the smallest number of (sub)permutation matrices. The Birkhoff-von Neumann decomposition theorem proves that there exists such a decomposition, but does not give a representation with the smallest number of permutation matrices. In particular, we consider the case when the optimal decomposition uses a constant number of matrices. We show that the problem is not fixed parameter tractable, and design a logarithmic approximation to the problem.