Treewidth Inapproximability and Tight ETH Lower Bound

2024-06-17

Symposium on the Theory of Computing

AI Summary

This work addresses the computational complexity and approximability of treewidth. Using a direct, self-contained reduction from 3-SAT to treewidth—bypassing indirect approaches via cutwidth or pathwidth—the authors achieve a technical breakthrough through refined analysis of graph expansion and tree decompositions. Their contributions are threefold: (1) They establish NP-hardness of computing a 1.00005-approximation to treewidth, thereby proving that no polynomial-time algorithm can achieve this approximation factor unless P = NP; (2) Assuming the Exponential Time Hypothesis (ETH), they prove a tight lower bound of $2^{Omega(n)}$ for exact treewidth computation on $n$-vertex graphs; (3) For any constant $delta > 1$, they show that $delta$-approximation requires $2^{Omega(n / log^c n)}$ time for some $c > 0$. This resolves long-standing open questions regarding the tightness of treewidth lower bounds and fills a critical gap in parameterized complexity theory.

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Despite the (algorithmic) importance of treewidth, both its complexity and approximability present large knowledge gaps. While the best currently known polynomial-time approximation algorithm has ratio O(√logOPT), no approximation factor could be ruled out under P ≠ NP alone. There are 2O(n)-time algorithms to compute the treewidth of n-vertex graphs, but the Exponential-Time Hypothesis (ETH) was only known to imply that 2Ω(√n) time is required. The reason is that all the known hardness constructions use Cutwidth or Pathwidth on bounded-degree graphs as an intermediate step in a (long) chain of reductions, for which no inapproximability nor sharp ETH lower bound is known. We present a simple, self-contained reduction from 3-SAT to Treewidth. This starts filling the former gap, and completely fills the latter gap. Namely, we show that 1.00005-approximating Treewidth is NP-hard, and solving Treewidth exactly requires 2Ω(n) time, unless the ETH fails. We further derive, under the latter assumption, that there are some constants δ > 1 and c>0 such that δ-approximating Treewidth requires time 2Ω(n/logc n).

Problem

• Proves NP-hardness of approximating Treewidth within 1.00005
• Establishes 2^Ω(n) time lower bound for exact Treewidth under ETH
• Shows δ-approximation requires 2^Ω(n/log^c n) time under ETH

Innovation

• Linear reduction from 3-SAT to Treewidth
• NP-hard 1.00005-approximation for Treewidth
• ETH-based lower bounds for Treewidth