Smoothed analysis of approximation algorithms
The paper by Bläser, Panagiotou, and Rao deals with concentration of the tour produced by Christofides' algorithm. No average-case approximation ratio is claimed, except for some experimental results.

The paper by Röglin and Vöcking (Math. Program., 2007) and an earlier paper by Beier and Vöcking (SIAM J. Comput., 2006) roughly state that smoothed polynomial time is equivalent to randomized pseudo-polynomial time. Here, pseudo-polynomial means running-time polynomial in the input size and the magnitude of the coefficients that are perturbed. This rules out smoothed polynomial complexity for strongly NP-hard optimization problems (unless NP=ZPP).

Concerning smoothed analysis and approximation, there are only very few papers that address specific problems or algorithms (Englert, Röglin, and Vöcking for the 2-opt heuristic for TSP; Bläser, Manthey, and Rao as well as Curticapean and Künnemann for partitioning heuristics; Karger and Onak for multi-dimensional packing). However, I am not aware of any structural connections between inapproximability and smoothed analysis.