Folder Complexes and Multiflow Combinatorial Dualities

In multiflow maximization problems, there are several combinatorial duality relations, such as Ford-Fulkerson’s max-flow min-cut theorem for single commodity flows, Hu’s max-biflow min-cut theorem for two-commodity flows, Lovász-Cherkassky duality theorem for free multiflows, and so on. In this paper, we provide a unified framework for such multiflow combinatorial dualities by using the notion of a folder complex, which is a certain 2-dimensional polyhedral complex introduced by Chepoi. We show that for a nonnegative weight μ on terminal set, μ-weighted maximum multiflow problem admits a combinatorial duality relation if and only if μ is represented by distances between certain subsets in a folder complex, and show that the corresponding combinatorial dual problem is a discrete location problem on the graph of the folder complex. This extends a result of Karzanov in the case of metric weights.