Multiflow and disjoint paths of minimum total cost
We discuss some earlier and recent results in the field of combinatorial network flow theory, considering problems on minimum cost maximum value multiflows (multicommodity flows), minimum cost maximum cardinality sets of edge-disjoint or openly disjoint paths, and related problems. Multiflows (multicommodity flows), minimum cost maximum cardinality sets of edge-disjoint or openly disjoint paths, and related problems. Throughout we deal with the undirected case. We exactly characterize the set of commodity graphs H for which, given an arbitrary network with nonnegative integer-valued capacities and costs, there exists an optimal multiflow f such that the denominator of each component of the vector f is bounded by a constant depending only on H. Moreover, for these H`s there are purely combinatorial polynomial time algorithms for finding such optimal solutions. Another, more complicated, problem is: given a graph G, a subset T of its vertices and a nonnegative function of cost on its edges, find a maximum cardinality set of pairwise edge-disjoint T-paths (i.e., paths in G connecting arbitrary pairs in T) such that the sum of costs of edges occurring in these paths is as small as possible. We give a minimax relation for this problem and develop a strongly polynomial algorithm to solve it. In fact, the former result generalizes the minimax relation involving the maximum number of edge-disjoint T-paths that has been established by Mader and, independently, Lomonosov. As a consequence, one can derive a description of the dominant polyhedron for the set of T,d-joins (a generalization of T-joins) and the dominant polyhedron for the set of so-called multi-joins of a graph. Finally, we present a minimax relation that concerns minimum cost maximum cardinality sets of pairwise openly disjoint T-paths, thus extending another result of Mader. The proof gives rise to a strongly polynomial solution algorithm.
- OSTI ID:
- 36191
- Report Number(s):
- CONF-9408161-; TRN: 94:009753-0473
- Resource Relation:
- Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.
- Country of Publication:
- United States
- Language:
- English
Similar Records
New network decomposition theorems with applications
Integral packing of trees and branchings