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.

翻译：我们证明了关于图中最大独立集近似问题的如下结果。非形式化地说，我们证明了任何具有"非平凡"近似比（作为输入图$G$顶点数的函数）的近似算法，都可以转化为一个达到几乎相同近似比（但作为$G$树宽的函数）的近似算法。更形式化地，我们证明对于任意函数$f$，若存在一个多项式时间的$(n/f(n))$-近似算法，则存在一个多项式时间的$O(tw \cdot\log{f(tw)}/f(tw))$-近似算法，其中$n$和$tw$分别表示输入图的顶点数和给定树分解的宽度。通过将我们的结果与Feige (2004)提出的目前最优的$O(n \cdot (\log \log n)^2/\log^3 n)$-近似算法进行流水线化，这便得到一个$O(tw \cdot (\log \log tw)^3/\log^3 tw)$-近似算法。