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 [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 $k_1,k_2$, 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 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 $P_1,P_2$ to the weak 2-linkage problem where each path $P_i$ has length at most $d(s_i,t_i)+c{\log }^{1+ϵ}n$ for some constant c.

有向图的弱2链接问题要求有给定的有向图和顶点$s_{\mathrm{1个}}，s_2，{Ť}_{\mathrm{1个}}，{Ť}_2$D是否包含一对不相交的弧形路径$P_{\mathrm{1个}}，P_2$ 这样 $P_{\mathrm{一世}}$ 是一个 $（s_{\mathrm{一世}}，{Ť}_{\mathrm{一世}}）$-路径。对于一般的有向图，这个问题是NP完全的，但是对于无环有向图，这个问题是多项式可解的[8]。最近显示[3]，如果D在圆弧上配备了一个权重函数w，它满足所有边都具有正权重，那么当两条路径都具有时，对于弱2连锁问题的变体，存在一个多项式算法是D中最短的路径。在本文中，我们考虑单位重量的情况，并证明对于每对常数${ķ}_{\mathrm{1个}}，{ķ}_2$，有一个多项式算法可确定输入图D是否具有一对弧不相交的路径$P_{\mathrm{1个}}，P_2$ 这样 $P_{\mathrm{一世}}$ 是一个 $（s_{\mathrm{一世}}，{Ť}_{\mathrm{一世}}）$-长度不超过 $d（s_{\mathrm{一世}}，{Ť}_{\mathrm{一世}}）+{ķ}_{\mathrm{一世}}$，对于 $\mathrm{一世}=\mathrm{1个}，2$，在哪里 $d（s_{\mathrm{一世}}，{Ť}_{\mathrm{一世}}）$ 表示最短的长度 $（s_{\mathrm{一世}}，{Ť}_{\mathrm{一世}}）$-路径。我们证明，除非指数时间假设（ETH）失败，否则没有多项式算法可确定解决方案的存在$P_{\mathrm{1个}}，P_2$到弱2链接问题，其中每条路径$P_{\mathrm{一世}}$ 最多有长度 $d（s_{\mathrm{一世}}，{Ť}_{\mathrm{一世}}）+C{\mathrm{日志}}^{\mathrm{1个}+ϵ}ñ$对于一些常数c。