Is there a example of Iterative Rounding in Approximation Algorithms for vertex weighted Graphs?
The simplest node-weighted network design problem is the node-weighted Steiner tree problem. Klein and Ravi considered this problem and obtained an $O(\log k)$-approximation (here $k$ is the number of terminals). They also showed that the Set Cover problem reduces to it in an approximation preserving fashion and hence we should not expect any better approximation. It is not hard to show that iterated rounding is not helpful for the  Set Cover problem. Consider the following example. We have $n$ elements $\{e_1,\ldots,e_n\}$ and $n$ sets $S_1,\ldots,S_n$ where $S_i$ contains all elements except $e_i$. If you look at the natural LP with a variable $x_i$ for set $S_i$ then a basic feasible optimal solution is to set $x_i = 1/(n-1)$ for each $i$. From this one can see that iterated rounding in the plain vanilla fashion won't give a good bound. Note that the integrality gap of the LP is $O(\log n)$. This example for Set Cover can be extended to the node-weighted Steiner tree problem.