Thin Tree Verification is coNP-Complete

Alice Moayyedi

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arXiv · 2512.25043

Thin Tree Verification is coNP-Complete

Alice Moayyedi

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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.

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Alice Moayyedi. 2025-12-31. Thin Tree Verification is coNP-Complete. https://arxiv.org/abs/2512.25043

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