Consider the Dominating Set problem in general graphs, and let $n$ be the number of vertices in a graph. A greedy approximation algorithm gives an approximation guarantee of factor $1 + \log n$, i.e. it's possible to find in polynomial-time a solution $S$ such that $|S| \leq (1 + \log n) opt$, where $opt$ is the size of a minimum dominating set. There are bounds showing that we cannot improve the dependency on $\log n$ much http://www.cs.duke.edu/courses/spring07/cps296.2/papers/p634-feige.pdf.
My question: is there an approximation algorithm which has a guarantee in terms of $opt$ instead of $n$? In graphs where $n$ is very large with respect to the optimum, a factor-$\log n$ approximation would be much worse than a factor $\log opt$ approximation. Is something like that known, or are there reasons why this cannot exist? I am happy with any polynomial-time algorithm which produces a solution $S$ such that $|S| \in O(opt^c)$ for some constant $c$.