A minimal completion of doubly substochastic matrix - Cao, Lei; Koyuncu, Selcuk; Parmer, Timmothy (Linear & multilinear algebra, 2016)
Let B be an doubly substochastic matrix and let s be the sum of all entries of B. In this paper, we show that B has a sub-defect of k,  which can be computed by taking the ceiling of if and only if there exists an doubly stochastic extension containing B as a submatrix and k minimal. We also propose a procedure constructing a minimal completion of B, and then express it as a convex combination of partial permutation matrices.