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Given a doubly-stochastic matrix $\pi$, Birkhoff-von Neumann theorem states that $\pi$ can be written as a convex combination of permutation matrices:

$$\pi=\sum_{i\leq k} \alpha_i B_i$$

where $\sum_{i \leq k} \alpha_i$ = 1 and $B_i$ are the permutation matrices. Such decomposition is generally not unique. I am looking for an algorithm that finds a particular decomposition that maximizes some objective. For example, suppose that for each $i$ there is a scalar $r_i$. I wish to find a decomposition s.t. $\sum_{i\leq k} \alpha_i r_i$ is maximized.

I would also be interested in an approximation algorithm or a heuristic.

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These authors try to minimize the number k of permutation matrices that are needed to decompose a specific DS matrix and show that this problem is NP-hard: https://hal.inria.fr/hal-01270331v2/document

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