Treedepth Inapproximability and Exponential ETH Lower Bound

2025-07-18

AI Summary

This paper investigates the computational and approximation complexity of the graph parameter *treedepth*. Addressing the unresolved hardness of exact computation and the unknown efficiency limits of approximation algorithms, the authors construct a fine-grained reduction from SAT to treedepth. They establish, for the first time, that (1.0003)-approximating treedepth is NP-hard; under the Exponential Time Hypothesis (ETH), computing treedepth exactly on an $n$-vertex graph requires $2^{Omega(n)}$ time, and any $(1+delta)$-approximation—for any absolute constant $delta > 0$—requires at least $2^{Omega(n/log^c n)}$ time. These results rule out the existence of a polynomial-time approximation scheme (PTAS) for treedepth and provide tight exponential lower bounds for both exact and approximate algorithms. Consequently, the work establishes fundamental theoretical barriers for algorithm design targeting structural graph parameters.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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 $O( ext{OPT} log^{3/2} ext{OPT})$-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 $ ext{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].

Problem

• Proves NP-hardness of 1.0003-approximation for treedepth
• Shows exact treedepth computation requires 2^Ω(n) time
• Establishes ETH-based lower bounds for approximation algorithms

Innovation

• Exact algorithm with $2^{O(k^2)} n$-time complexity
• Polynomial-time $O( ext{OPT} log^{3/2} ext{OPT})$-approximation
• Direct reduction from Satisfiability to Treedepth