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arXiv · 1006.5520

On duality and fractionality of multicommodity flows in directed networks

Hiroshi Hirai·Shungo Koichi
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Abstract

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.

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Hiroshi Hirai, Shungo Koichi. 2010-06-29. On duality and fractionality of multicommodity flows in directed networks. https://arxiv.org/abs/1006.5520

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