Linkages in Digraphs

Abstract

We saw in Chapter 5 that it is easy to check (e.g., using flows) whether a directed multigraph D=(V,A) has k (arc)-disjoint paths P₁,P₂,…,P _k from a subset X⊂V to another subset Y⊂V and we can also find such paths efficiently. On many occasions (e.g., in practical applications) we need to be able to specify the initial and terminal vertices of each P _i , i=1,2,…,k, that is, we wish to find a so-called linkage from X={x₁,x₂,…,x _k } to Y={y₁,y₂,…,y _k } such that P _i is an (x _i ,y _i )-path for every i∈[k]. This problem is considerably more difficult and is in fact $\mathcal{NP}$-complete already when k=2. In this chapter we start by giving a proof of this fact and then we discuss a number of results on sufficient conditions for the existence of linkages, polynomial algorithms for special classes of digraphs, including acyclic, planar and semicomplete digraphs in the case of vertex disjoint paths and acyclic digraphs and some generalizations of tournaments in the case of arc-disjoint paths. The reader will see that quite a lot can be said about the linkage problems for special classes of digraphs and that still the problems are not trivial for these classes of digraphs. Finally we briefly discuss topics such as multi commodity flows and subdivisions of transitive tournaments in digraphs with large out-degree.