Theoretical Computer Science ( IF 1.1 ) Pub Date : 2020-01-15 , DOI: 10.1016/j.tcs.2020.01.012 J. Bang-Jensen , T. Bellitto , W. Lochet , A. Yeo
The weak 2-linkage problem for digraphs asks for a given digraph and vertices whether D contains a pair of arc-disjoint paths such that is an -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 , there is a polynomial algorithm which decides whether the input digraph D has a pair of arc-disjoint paths such that is an -path of length no more than , for , where denotes the length of the shortest -path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution to the weak 2-linkage problem where each path has length at most for some constant c.
中文翻译:
具有长度约束的有向2链问题
有向图的弱2链接问题要求有给定的有向图和顶点D是否包含一对不相交的弧形路径 这样 是一个 -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,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 这样 是一个 -长度不超过 ,对于 ,在哪里 表示最短的长度 -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到弱2链接问题,其中每条路径 最多有长度 对于一些常数c。