# The directed 2-linkage problem with length constraints

The weak 2-linkage problem for digraphs asks for a given digraph and vertices s_1,s_2,t_1,t_2 whether D contains a pair of arc-disjoint paths P_1,P_2 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 <cit.>. Recently it was shown <cit.> 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 constants k_1,k_2, there is a polynomial algorithm which decides whether the input digraph D has a pair of arc-disjoint paths P_1,P_2 such that P_i is an (s_i,t_i)-path and the length of P_i is 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 P_1,P_2 to the weak 2-linkage problem where each path P_i has length at most d(s_i,t_i)+ clog^1+ϵn for some constant c. We also prove that the weak 2-linkage problem remains NP-complete if we require one of the two paths to be a shortest path while the other path has no restriction on the length.

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