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Multiflows and disjoint paths of minimum total cost

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Abstract

In this paper we discuss a number of recent and earlier results in the field of combinatorial optimization that concerns problems on minimum cost multiflows (multicommodity flows) and edge-disjoint paths. More precisely, we deal with an undirected networkN consisting of a supply graphG, a commodity graphH and nonnegative integer-valued functions of capacities and costs of edges ofG, and consider the problems of minimizing the total cost among (i) all maximum multiflows, and (ii) all maximuminteger multiflows.

For problem (i), we discuss the denominators behavior in terms ofH. The main result is that ifH is complete (i.e. flows between any two terminals are allowed) then (i) has ahalf-integer optimal solution. Moreover, there are polynomial algorithms to find such a solution. For problem (ii), we give an explicit combinatorial minimax relation in case ofH complete. This generalizes a minimax relation, due to Mader and, independently, Lomonosov, for the maximum number of edge-disjoint paths whose ends belong to a prescribed subset of nodes of a graph. Also there exists a polynomial algorithm when the capacities are all-unit.

The minimax relation for (ii) withH complete allows to describe the dominant for the set of (T, d)-joins (extending the notion ofT-join) and the dominant for the set of maximum multi-joins of a graph. Also other relevant results are reviewed and open questions are raised. © 1997 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

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Authors and Affiliations

  1. Institute for System Analysis of Russian Acad. Sci., 9, Prospect 60 Let Oktyabrya, 117312, Moscow, Russia

    Alexander V. Karzanov

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  1. Alexander V. Karzanov
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This research was supported in part by European Union grant INTAS-93-2530.

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Karzanov, A.V. Multiflows and disjoint paths of minimum total cost. Mathematical Programming 78, 219–242 (1997). https://doi.org/10.1007/BF02614372

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