We study a generalization of the well-known disjoint paths problem which we call the metric Menger problem, denoted MM(r,k), where one is given two subsets of a graph and must decide whether they can be connected by $k$ paths of pairwise distance at least $r$. We prove that this problem is NP-complete for every $r\geq 3$ and $k\geq 2$ by giving a reduction from 3SAT. This resolves a conjecture recently stated by Georgakopoulos and Papasoglu. On the other hand, we show that the problem is in XP when parameterised by treewidth and maximum degree by observing that it is `locally checkable'. In the case $r\leq 3$, we prove that it suffices to parameterise by treewidth. We also state some open questions relating to this work.

翻译：我们研究了著名的不相交路径问题的一个推广，称为度量Menger问题，记为MM(r,k)，其中给定一个图的两个子集，需要判断它们是否可以通过$k$条相互距离至少为$r$的路径相连。通过从3SAT进行归约，我们证明该问题对每个$r\geq 3$和$k\geq 2$都是NP完全的，从而解决了Georgakopoulos和Papasoglu最近提出的一个猜想。另一方面，通过观察到该问题是“局部可检查的”，我们证明当以树宽和最大度作为参数时，该问题属于XP类。在$r\leq 3$的情况下，我们证明仅以树宽作为参数就足够了。同时，我们还陈述了与本研究相关的一些开放问题。