Abstract

In this paper, we introduce the upgrading problem of edge-disjoint paths. In the off-line upgrading problem, a supply graph G with integer capacities and two demand graphs H 1 and H 2 with unit demands are given on the same vertex set. Our task is to determine the maximum size of a set F ⊆ E ( H 1 ) ∩ E ( H 2 ) such that F has an integer routing in G which can be extended both to an integer routing of H 1 and to an integer routing of H 2 . In the online upgrading problem, we are given a supply graph G with integer capacities, a demand graph H with an integer routing, and another demand graph H 2 with unit demands such that E ( H ) ⊆ E ( H 2 ) . Our task is to determine the maximum size of a set F ⊆ E ( H ) such that the restriction of the given routing to F can be extended to an integer routing of H 2 . Thus, depending on whether the graphs are directed or undirected, we have four different versions. We give algorithmic proofs of minimax formulas for the case when G is a ring and the demand graphs are stars with the same center. All four versions are NP-complete for general graphs.

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