On duality and fractionality of multicommodity flows in directed networks

In this paper we address a topological approach to multiflow (multicommodity flow) problems in directed networks. Given a terminal weight $μ$, we define a metrized polyhedral complex, called the directed tight span $T_μ$, and prove that the dual of $μ$-weighted maximum multiflow problem reduces to a facility location problem on $T_μ$. Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by $μ$. 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.

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