  The problem of determining $DS_n$, the complex numbers that occur as an
eigenvalue of an $n$-by-$n$ doubly stochastic matrix, has been a target of
study for some time. The Perfect-Mirsky region, $PM_n$, is contained in $DS_n$,
and is known to be exactly $DS_n$ for $n \leq 4$, but strictly contained within
$DS_n$ for $n = 5$. Here, we present a Boundary Conjecture that asserts that
the boundary of $DS_n$ is achieved by eigenvalues of convex combinations of
pairs of (or single) permutation matrices. We present a method to efficiently
compute a portion of $DS_n$, and obtain computational results that support the
Boundary Conjecture. We also give evidence that $DS_n$ is equal to $PM_n$ for
certain $n > 5$.
