Foregger's Conjecture on Minimum Permanents of Nearly Decomposable Doubly Stochastic Matrices

We prove Foregger’s conjecture: every nearly decomposable doubly stochastic matrix A of order n ≥ 2 satisfies per A ≥ 21−n. Equality holds exactly for matrices obtained from ½(In + Pn) by independent row and column permutations, where Pn is a single-cycle permutation matrix. In fact, for every fixed nearly decomposable support whose bipartite graph is not a single cycle, the minimum permanent on its closed doubly stochastic face is strictly larger than 21−n.