  For a partially specified stochastic matrix, we consider the problem of
completing it so as to minimize Kemeny's constant. We prove that for any
partially specified stochastic matrix for which the problem is well-defined,
there is a minimizing completion that is as sparse as possible. We also find
the minimum value of Kemeny's constant in two special cases: when the diagonal
has been specified, and when all specified entries lie in a common row.
