Variants of Densest Subgraph Problems
This is meant more as a comment than as a solution, but is a bit too long to fit in the comment box. There is a bit of handwaving and ignoring of small factors, but I can formalize the work below further if requested.

Here's a heuristic reason why (a) might be difficult to approximate within a small factor. 

Consider the random-in-random PLANTED DENSE SUBGRAPH problem, which one to reliably distinguish between the Erdos-Renyi graph $G(n,p)$ and the graph $G(n,p)$ with a planted $G(n',p')$ for certain values of $n,n',p,p'$. It is a difficult problem -- conjectured hard by some -- to do this in polynomial time for $p = \frac{1}{n^{1/2}}$, $n' = n^{1/2}$, $p' = \frac{1}{n^{1/4 + \epsilon}}$. That is, it's hard to distinguish a "pure" $G(n,1/\sqrt{n})$ graph (where each induced subgraph of size $\sqrt{n}$ has average degree near $1$) from instances where some subset of $\sqrt{n}$ vertices has so many edges planted between them that the average degree is roughly $n^{1/4 - \epsilon}$. 

In such instances, there's quite a large gap in the maximum edge densities in subgraphs of size at least $\sqrt{n}$. In particular, while the planted portion of the planted-instance graph has edge density roughly $\frac{1}{n^{1/4+\epsilon}}$, just about any size $\sqrt{n}$ (or higher) subgraph of the non-planted $G(n,1/\sqrt{n})$ graph will have edge density about $1/\sqrt{n}$, a gap of roughly $n^{1/4 - \epsilon}$. 

Any approximation algorithm with a stronger guarantee than $n^{1/4 - \epsilon}$ can thus be used to solve a conjectured-hard class of instances of Planted-DkS.