Folder Complexes and Multiflow Combinatorial Dualities
Hiroshi Hirai · SIAM Journal on Discrete Mathematics · 2011
In multiflow maximization problems, there are several combinatorial duality relations, such as the Ford–Fulkerson max-flow min-cut theorem for single commodity flows, Hu’s max-biflow min-cut theorem for two-commodity flows, the 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 [Formula: see text] on terminal set, the [Formula: see text]-weighted maximum multiflow problem admits a combinatorial duality relation if and only if [Formula: see text] is represented by distances between certain subsets in a folder complex, and we 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.