On return path packing

We consider a path packing problem: given a supply graph G with a node-set N and a demand graph (T,S) with T⊆N, find the maximal number of edge-disjoint paths in G whose end-pairs belong to S; the network (G,T) is assumed to be Eulerian. Karzanov’s condition on cliques of the complementary graph (T,S) (Polyhedra related to undirected multicommodity flows, Linear Algebra and its Applications 114/115 (1989) 293) appreciably restricts the class of such problems. The excluded cases are all known to be NP-hard, while the retained problems, except those related to the cut condition, are still open. The paper presents a max–min theorem for the easiest of these problems, with (T,S) isomorphic to K2,r, r>2. The method implements an approach of “smooth relaxation” implicitly developed in prior research in the area. The proof is nonconstructive; the algorithmic aspect of the problem is still open.