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.

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