Inapproximability of set cover: can I assume m=poly(n)? I am trying to show that a certain problem is inapproximable by a reduction from set cover. My reduction transforms an instance with ground set of size $n$ and $m$ sets into an instance of my problem where a certain parameter $r$ is of size $O(n+m)$. I can then show that an instance of set cover where the cover size is s corresponds to an instance of my problem where the size of the optimal solution is $2s$ (or something like this), and vice versa. I would like to invoke Raz-Safra to conclude that my problem is inapproximable up to a factor of $c  \log{r}$, for some constant $c$. This would work fine if I could assume that $m$ is bounded by a fixed polynomial of $n$. Does anyone know if it is kosher to assume this? This is certainly true for the family of instances used in the standard NP-hardness proof for set cover, but I am not sure if this remains the case for the kind of PCP reductions employed by Raz and Safra.
Yes, the number of sets m in a set-cover instance is polynomial in the number of elements.
By the way -- the state of the art hardness results for Set-Cover are:

With Noga Alon and Muli Safra, we showed how to use the Raz-Safra/Arora-Sudan PCP to get a better constant $c$ in the hardness factor $c\log n$.
http://people.csail.mit.edu/dmoshkov/papers/k-restrictions/k-rest-full.ps

Feige showed how to get the optimal hardness factor $(1-\epsilon)\ln n$, assuming $NP\not\subseteq DTIME(n^{\log\log n})$.
http://www.cs.duke.edu/courses/spring07/cps296.2/papers/p634-feige.pdf

I recently published a note on how to adapt Feige's reduction to an NP-hardness result (i.e., a result based on $P\neq NP$), assuming a plausible conjecture about PCPs (A conjecture I call &quot;The Projection Games Conjecture&quot; - a specialization of the 1993 &quot;Sliding Scale Conjecture&quot; to projection games).
http://eccc.hpi-web.de/report/2011/112/
(I later found out that the reduction gives an optimal tradeoff between $\epsilon$ and the reduction blow-up).