On duality and fractionality of multicommodity flows in directed networks

Authors:
Hiroshi Hirai;Shungo Koichi
Affiliations:
Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan;Department of Systems Design and Engineering, Nanzan University, Seto 489-0863, Japan
Venue:
Discrete Optimization
Year:
2011

Citing 3
Cited 0

Theory of linear and integer programming

Theory of linear and integer programming
Minimum 0-extensions of graph metrics

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Tight spans of distances and the dual fractionality of undirected multiflow problems

Journal of Combinatorial Theory Series B

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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 @m, we define a metrized polyhedral complex, called the directed tight span T"@m, and prove that the dual of the @m-weighted maximum multiflow problem reduces to a facility location problem on T"@m. Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by @m. 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.