Thin Tree Verification is coNP-Complete

TL;DR

This paper proves that verifying whether a given spanning tree is $oldsymbol{ extit{ ext{α}}}$-thin in a graph is coNP-hard, highlighting the computational difficulty of the problem and impacting approaches to related conjectures and algorithms.

Contribution

It establishes the coNP-hardness of verifying the thinness of a tree, a problem previously unresolved, affecting research on the Thin Tree Conjecture and approximation algorithms for ATSP.

Findings

Verifying α-thinness of a tree is coNP-hard.

Implications for the Thin Tree Conjecture and related algorithms.

Highlights computational complexity of a key verification problem.

Abstract

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