  Generalizing the bit thread formalism, we use convex duality to derive dual
flow programs to the bipartite and multipartite holographic entanglement of
purification proposals and then prove several inequalities using these
constructions. In the multipartite case we find the flows exhibit novel
behavior which allows for a constrained flux on the boundary of the homology
region. We show this flux can be made distinct from bipartite terms and
reflects the truly multipartite portion of the holographic entanglement of
purification.
