NP-complete problems with optimal approximation in poly-time
One example is the chromatic index (edge-coloring number) of undirected graphs. By Vizing's theorem, the chromatic index either equals the max-degree $\Delta$ or equals $\Delta+1$. 


Vizing's proof gives a polynomial time (inductive) procedure for finding a coloring with $\Delta+1$ colors. 
Holyer has proved that deciding whether the chromatic index equals $\Delta$ is NP-complete.