Treedepth Inapproximability and Exponential ETH Lower Bound
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916849860476928 |
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| author | Bonnet, Édouard Neuen, Daniel Sokołowski, Marek |
| author_facet | Bonnet, Édouard Neuen, Daniel Sokołowski, Marek |
| contents | Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a $2^{O(k^2)} n$-time exact algorithm and a polynomial-time $O(\text{OPT} \log^{3/2} \text{OPT})$-approximation algorithm, where the former algorithm returns an elimination forest of height $k$ (witnessing that treedepth is at most $k$) for the $n$-vertex input graph $G$, or correctly reports that $G$ has treedepth larger than $k$, and $\text{OPT}$ is the actual value of the treedepth. On the complexity side, exactly computing treedepth is NP-complete, but the known reductions do not rule out a polynomial-time approximation scheme (PTAS), and under the Exponential Time Hypothesis (ETH) only exclude a running time of $2^{o(\sqrt n)}$ for exact algorithms.
We show that 1.0003-approximating treedepth is NP-hard, and that exactly computing the treedepth of an $n$-vertex graph requires time $2^{Ω(n)}$, unless the ETH fails. We further derive that there exist absolute constants $δ, c > 0$ such that any $(1+δ)$-approximation algorithm requires time $2^{Ω(n / \log^c n)}$. We do so via a simple direct reduction from Satisfiability to Treedepth, inspired by a reduction recently designed for Treewidth [STOC '25]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13818 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Treedepth Inapproximability and Exponential ETH Lower Bound Bonnet, Édouard Neuen, Daniel Sokołowski, Marek Computational Complexity Data Structures and Algorithms Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a $2^{O(k^2)} n$-time exact algorithm and a polynomial-time $O(\text{OPT} \log^{3/2} \text{OPT})$-approximation algorithm, where the former algorithm returns an elimination forest of height $k$ (witnessing that treedepth is at most $k$) for the $n$-vertex input graph $G$, or correctly reports that $G$ has treedepth larger than $k$, and $\text{OPT}$ is the actual value of the treedepth. On the complexity side, exactly computing treedepth is NP-complete, but the known reductions do not rule out a polynomial-time approximation scheme (PTAS), and under the Exponential Time Hypothesis (ETH) only exclude a running time of $2^{o(\sqrt n)}$ for exact algorithms. We show that 1.0003-approximating treedepth is NP-hard, and that exactly computing the treedepth of an $n$-vertex graph requires time $2^{Ω(n)}$, unless the ETH fails. We further derive that there exist absolute constants $δ, c > 0$ such that any $(1+δ)$-approximation algorithm requires time $2^{Ω(n / \log^c n)}$. We do so via a simple direct reduction from Satisfiability to Treedepth, inspired by a reduction recently designed for Treewidth [STOC '25]. |
| title | Treedepth Inapproximability and Exponential ETH Lower Bound |
| topic | Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2507.13818 |