Half-integral linkages in highly connected directed graphs*

We study the half-integral k-Directed Disjoint Paths Problem (1/2 kDDPP) in highly strongly connected digraphs. The integral kDDPP is NP-complete even when restricted to instances where k = 2, and the input graph is L-strongly connected, for any L > 1. We show that when the integrality condition is relaxed to allow each vertex to be used in two paths, the problem becomes efficiently solvable in highly connected digraphs (even with k as part of the input). Specifically, we show that there is an absolute constant c such that for each k>2 there exists L(k) such that ½ kDDPP is solvable in time O(jV(G)jc) for a L(k)-strongly connected directed graph G. As the function L(k) grows rather quickly, we also show that 1/2kDDPP is solvable in time O(jV(G)j f (k)) in (36k3 +2k)-strongly connected directed graphs. We also show that for each e 1 deciding half-integral feasibility of kDDPP instances is NP-complete when k is given as part of the input, even when restricted to graphs with strong connectivity ek.