The directed 2-linkage problem with length constraints

The WEAK 2-LINKAGE problem for digraphs asks for a given digraph and vertices s₁,s₂,t₁,t₂ whether D contains a pair of arc-disjoint paths P₁,P₂ such that P_i is an (s_i,t_i)-path. This problem is NP-complete for general digraphs but polynomially solvable for acyclic digraphs [8]. Recently it was shown [3] that if D is equipped with a weight function w on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak-2-linkage problem when both paths have to be shortest paths in D. In this paper we consider the unit weight case and prove that for every pair of constants k₁,k₂, there is a polynomial algorithm which decides whether the input digraph D has a pair of arc-disjoint paths P₁,P₂ such that P_i is an (s_i,t_i)-path of length no more than d(s_i,t_i)+k_i, for i=1,2, where d(s_i,t_i) denotes the length of the shortest (s_i,t_i)-path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution P₁,P₂ to the WEAK 2-LINKAGE problem where each path P_i has length at most d(s_i,t_i)+clog^1+ϵ⁡n for some constant c.