Maximum Multiflow Problems Research Articles

The Maximum Multiflow Problems with Bounded Fractionality

We consider the weighted maximum multiflow problem with respect to a terminal weight. We show that if the dimension of the tight span associated with the weight is at most 2, then this problem has a 1/12-integral optimal multiflow for every Eulerian supply graph. This result solves a weighted generalization of Karzanov's conjecture for classifying commodity graphs with finite fractionality. In addition, our proof technique proves the existence of an integral or half-integrality optimal multiflow for a large class of multiflow maximization problems, and it gives a polynomial time algorithm.

On duality and fractionality of multicommodity flows in directed networks

In this paper we address a topological approach to multiflow (multicommodity flow) problems in directed networks. Given a terminal weight μ , we define a metrized polyhedral complex, called the directed tight span T μ , and prove that the dual of the μ -weighted maximum multiflow problem reduces to a facility location problem on T μ . Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by μ . 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.

Folder Complexes and Multiflow Combinatorial Dualities

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 Lovasz–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, the 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 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.

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

In this paper, we give a complete characterization of the class of weighted maximum multiflow problems whose dual polyhedra have bounded fractionality. This is a common generalization of two fundamental results of Karzanov. The first one is a characterization of commodity graphs H for which the dual of maximum multiflow problem with respect to H has bounded fractionality, and the second one is a characterization of metrics d on terminals for which the dual of metric-weighed maximum multiflow problem has bounded fractionality. A key ingredient of the present paper is a nonmetric generalization of the tight span, which was originally introduced for metrics by Isbell and Dress. A theory of nonmetric tight spans provides a unified duality framework to the weighted maximum multiflow problems, and gives a unified interpretation of combinatorial dual solutions of several known min–max theorems in the multiflow theory.

On one maximum multiflow problem and related metrics

We consider the undirected maximum multiflow (multicommodity flow) problem in the case when the commodity graph is the disjoint union of K 3 and K 2. We prove that if the supply graph satisfies a certain Eulerian-type condition, then the problem has an integer optimal solution. To obtain this result, we first study the corresponding dual problem on metrics and show that an optimal solution to the latter is achieved on some (2,3)-metric or some 3-cut metric.