Tight Inapproximability of Max Independent Set in Triangle-Free Graphs

2026-08-06

AI Summary

This work investigates the approximability of the Maximum Independent Set problem in triangle-free graphs. By combining the Moser–Tardos resampling algorithm with the probabilistic analysis technique of Haeupler, Saha, and Srinivasan, the authors constructively prove that, for any ε > 0, no polynomial-time algorithm can achieve an n^{1/2−ε}-approximation unless NP ⊆ BPP. This result closes the long-standing gap between the known upper and lower bounds on the approximation ratio for this problem in triangle-free graphs. Furthermore, the approach is extended to graphs excluding any fixed family of cycle-containing forbidden subgraphs, thereby establishing a unified framework for inapproximability results across a broad class of sparse graph families.

Abstract

For every $\varepsilon > 0$, it is NP-hard to $n^{1-\varepsilon}$-approximate Max Independent Set in $n$-vertex graphs [Hastad '96, Zuckerman '07]. In triangle-free graphs, a simple argument gives a polynomial-time $n^{1/2}$-approximation algorithm, whereas, for every $\varepsilon > 0$, an $n^{1/4-\varepsilon}$-approximation algorithm would imply that NP $\subseteq$ BPP [Bonnet, Thomassé, Tran, Watrigant; ESA '20]. In this note, we close this gap by proving the corresponding hardness against $n^{1/2-\varepsilon}$-approximation algorithms. The reduction is very simple and uses the Moser-Tardos resampling algorithm to make the constructed graphs triangle-free. The soundness uses a result of Haeupler, Saha, and Srinivasan building on the proof of Moser and Tardos, to upper-bound the probability that a fixed relatively large subset is an independent set after the Moser-Tardos algorithm terminates. We generalize this scheme and show that, for any nonempty finite family $\mathcal F$ of graphs, each containing at least one cycle, for any $\varepsilon > 0$, an $n^{μ(\mathcal F)-\varepsilon}$-approximation algorithm for Max Independent Set in graphs excluding every member of $\mathcal F$ as a subgraph implies that NP $\subseteq$ BPP, where $μ(\mathcal F) := 1 - \max\limits_{H \in \mathcal F}~\min\limits_{U \subseteq V(H), H[U] \text{contains a cycle}} (|U|-2)/(|E(H[U])|-1)$.

Problem

• Max Independent Set
• Triangle-Free Graphs
• Inapproximability
• Hardness of Approximation
• Graph Exclusion

Innovation

• inapproximability
• triangle-free graphs
• Moser-Tardos algorithm
• maximum independent set
• hardness of approximation