Tight Inapproximability of Max Independent Set in Triangle-Free Graphs
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, Thomass\'e, 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 $n^{\mu(\mathcal F)-\varepsilon}$-approximation algorithm for Max Independent Set in graphs excluding every member of $\mathcal F$ as a subgraph implies that NP $\subseteq$ BPP, where $\mu(\mathcal F) := 1 - \max\limits_{H \in \mathcal F}~\min\limits_{U \subseteq V(H), H[U] \text{contains a cycle}} (|U|-2)/(|E(H[U])|-1)$.