On undirected two‐commodity integral flow, disjoint paths and strict terminal connection problems

Abstract

Even, Itai, and Shamir (1976) proved simple two‐commodity integral flow is NP‐complete both in the directed and undirected cases. In particular, the directed case was shown to be NP‐complete even if one demand is unitary, which was improved by Fortune, Hopcroft and Wyllie (1980) who proved the problem is still NP‐complete if both demands are unitary. The undirected case, on the other hand, was proved by Robertson and Seymour (1995) to be polynomial‐time solvable if both demands are constant. Nevertheless, the complexity of the undirected case with exactly one constant demand has remained unknown. We close this 40‐year complexity gap, by showing the undirected case is NP‐complete even if exactly one demand is unitary. As a by product, we obtain the NP‐completeness of determining whether a graph contains 1 + d pairwise vertex‐disjoint paths, such that one path is between a given pair of vertices and d paths are between a second given pair of vertices. Additionally, we investigate the complexity of another related network design problem called strict terminal connection.

The Strict Terminal Connection Problem on Chordal Bipartite Graphs

“…Table 1 summarises the known complexity results of S-TCP with respect to the parameters ℓ, r, ∆(G), and the classes of split graphs and cographs. In addition to these results, in [4], S-TCP was studied from the perspective of disjoint paths and integral commodity flow problems.…”

Preface: Special issue on network analytics and optimization

“…De Melo et al [3] study the complexity of a simple two‐commodity integral flow problem. They prove that the problem is NP‐hard on undirected graphs even if one of the demands is unitary.…”