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 μμ, we define a metrized polyhedral complex, called the directed tight span TμTμ, and prove that the dual of the μμ-weighted maximum multiflow problem reduces to a facility location problem on TμTμ. Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by μμ. By utilizing this duality, we establish the classifications of terminal weights admitting a combinatorial min–max relation (i) for every network and (ii) for every Eulerian network. Our result includes the Lomonosov–Frank theorem for directed free multiflows and Ibaraki–Karzanov–Nagamochi’s directed multiflow locking theorem as special cases.