On duality and fractionality of multicommodity flows in directed networks

TL;DR

This paper introduces a topological framework for multicommodity flows in directed networks, linking duality to facility location problems on specialized polyhedral complexes, and classifies terminal weights with min-max relations.

Contribution

It develops a topological approach using the directed tight span and tropical polytope to analyze duality and fractional properties of multicommodity flows in directed networks.

Findings

Dual of maximum multiflow reduces to a facility location problem on the directed tight span.

In Eulerian networks, the problem reduces further to a tropical polytope.

Classifies terminal weights that admit min-max relations for all networks and Eulerian networks.

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.