Half-integral linkages in highly connected directed graphs

We study the half-integral k-Directed Disjoint Paths Problem (12 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 1 2kDDPP is solvable in time O(|V (G)|c) for a L(k)-strongly connected directed graph G. As the function L(k) grows rather quickly, we also show that 12 kDDPP is solvable in time O(|V (G)|f(k)) in (36k3 + 2k)-strongly connected directed graphs. We show that for each ϵ < 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 ϵ,k.