  We consider the problem of how to fairly distribute the minimum sum-rate
among the users in communication for omniscience (CO). We formulate a problem
of minimizing a weighted quadratic function over a submodular base polyhedron
which contains all achievable rate vectors, or transmission strategies, for CO
that have the same sum-rate. By solving it, we can determine the rate vector
that optimizes the Jain's fairness measure, a more commonly used fairness index
than the Shapley value in communications engineering. We show that the
optimizer is a lexicographically optimal (lex-optimal) base and can be
determined by a decomposition algorithm (DA) that is based on submodular
function minimization (SFM) algorithm and completes in strongly polynomial
time. We prove that the lex-optimal minimum sum-rate strategy for CO can be
determined by finding the lex-optimal base in each user subset in the
fundamental partition and the complexity can be reduced accordingly.
