An $α$-thin tree $T$ of a graph $G$ is a spanning tree such that every cut of $G$ has at most an $α$ proportion of its edges in $T$. The Thin Tree Conjecture proposes that there exists a function $f$ such that for any $α> 0$, every $f(α)$-edge-connected graph has an $α$-thin tree. Aside from its independent interest, an algorithm which could efficiently construct an $O(1)/k$-thin tree for a given $k$-edge-connected graph would directly lead to an $O(1)$-approximation algorithm for the asymmetric travelling salesman problem (ATSP)(arXiv:0909.2849). However, it was not even known whether it is possible to efficiently verify that a given tree is $α$-thin. We prove that determining the thinness of a tree is coNP-hard.

翻译：图$G$的一棵$α$-细树$T$是指一棵生成树，使得$G$的任意割中至多有$α$比例的边位于$T$中。细树猜想提出，存在一个函数$f$，使得对于任意$α>0$，每个$f(α)$-边连通图都有一棵$α$-细树。除了其独立的理论意义外，对于一个给定的$k$-边连通图，能够高效构造一棵$O(1)/k$-细树的算法将直接导向非对称旅行商问题（ATSP）的$O(1)$-近似算法（arXiv:0909.2849）。然而，此前甚至不清楚是否能够高效验证给定的一棵树是否为$α$-细树。我们证明了判定一棵树的细度是coNP困难的。