Abstract
Generalizing the two-commodity flow theorem of Rothschild and Whinston [Oper. Res., 14 (1966), pp. 377--387] and the multiflow theorem of Lovasz [Acta Mat. Akad. Sci. Hungaricae, 28 (1976), pp. 129--138] and Cherkasky [Ekonom.-Mat. Metody, 13 (1977), pp. 143--151], Karzanov and Lomonosov [Mathematical Programming, O. I. Larichev, ed., Institute for System Studies, 1978, pp. 59--66] in 1978 proved a min-max theorem on maximum multiflows. Their original proof is quite long and technical and relies on earlier investigations into metrics. The main purpose of the present paper is to provide a relatively simple proof of this theorem. Our proof relies on the locking theorem, which is another result of Karzanov and Lomonosov, and the polymatroid intersection theorem of Edmonds [Combinatorial Structures and Their Applications, R. Guy, H. Hanani, N. Sauer, and J. Schonheim, eds., Gordon and Breach, 1970, pp. 69--87]. For completeness, we also provide a simplified proof of the locking theorem. Finally, we introduce the notion of a node demand problem and, as another application of the locking theorem, we derive a feasibility theorem concerning it. The presented approach gives rise to (combinatorial) polynomial-time algorithms.
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