  In this paper we address a topological approach to multiflow (multicommodity
flow) problems in directed networks. Given a terminal weight $\mu$, we define a
metrized polyhedral complex, called the directed tight span $T_{\mu}$, and
prove that the dual of $\mu$-weighted maximum multiflow problem reduces to a
facility location problem on $T_{\mu}$. Also, in case where the network is
Eulerian, it further reduces to a facility location problem on the tropical
polytope spanned by $\mu$. By utilizing this duality, we establish the
classifications of terminal weights admitting combinatorial min-max relation
(i) for every network and (ii) for every Eulerian network. Our result includes
Lomonosov-Frank theorem for directed free multiflows and
Ibaraki-Karzanov-Nagamochi's directed multiflow locking theorem as special
cases.
