Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs

We study the inapproximability of Vertex Cover and Independent Set on degree $d$ graphs. We prove that: \begin{itemize} \item Vertex Cover is Unique Games-hard to approximate to within a factor $2 - (2+o_d(1)) \frac{ \log\log d}{ \log d}$. This exactly matches the algorithmic result of Halperin \cite{halperin02improved} up to the $o_d(1)$ term. \item Independent Set is Unique Games-hard to approximate to within a factor $O(\frac{d}{\log^2 d})$. This improves the $\frac{d}{\log^{O(1)}(d)}$ Unique Games hardness result of Samorodnitsky and Trevisan \cite{samorodnitsky06gowers}. Additionally, our result does not rely on the construction of a query efficient PCP as in \cite{samorodnitsky06gowers}. \end{itemize}