Thin Tree Verification is coNP-Complete

Abstract

An $\hat{I}\pm$-thin tree $T$ of a graph $G$ is a spanning tree such that every cut of $G$ has at most an $\hat{I}\pm$ proportion of its edges in $T$. The Thin Tree Conjecture proposes that there exists a function $f$ such that for any $\hat{I}\pm >0$, every $f(\hat{I}\pm )$-edge-connected graph has an $\hat{I}\pm$-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 $\hat{I}\pm$-thin. We prove that determining the thinness of a tree is coNP-hard.

8 pages, 1 figure