Minimal number of permutations in Birkhoff decompositions
Determine tight lower bounds and, when possible, exact minimal cardinalities for the number of permutation matrices required to express a given n×n doubly stochastic matrix as a convex combination of permutation matrices, ideally characterized in terms of matrix size, sparsity, and structure.
References
It is possible to obtain tighter upper bounds on the number of permutations required depending on the number of zeros in the doubly stochastic matrix (see, e.g., [Brualdi_1982]); however, the lower bound, i.e., the minimum number of permutations matrices required, is generally not known.
— Quantum Optimization Benchmark Library -- The Intractable Decathlon
(Koch et al., 4 Apr 2025) in Section 4.3 (Minimum Birkhoff Decomposition), Background