  We prove the following result about approximating the maximum independent set
in a graph. Informally, we show that any approximation algorithm with a
``non-trivial'' approximation ratio (as a function of the number of vertices of
the input graph $G$) can be turned into an approximation algorithm achieving
almost the same ratio, albeit as a function of the treewidth of $G$. More
formally, we prove that for any function $f$, the existence of a polynomial
time $(n/f(n))$-approximation algorithm yields the existence of a polynomial
time $O(tw \cdot\log{f(tw)}/f(tw))$-approximation algorithm, where $n$ and $tw$
denote the number of vertices and the width of a given tree decomposition of
the input graph. By pipelining our result with the state-of-the-art $O(n \cdot
(\log \log n)^2/\log^3 n)$-approximation algorithm by Feige (2004), this
implies an $O(tw \cdot (\log \log tw)^3/\log^3 tw)$-approximation algorithm.
