If there is an polynomial time approximation to an NP-complete problem, is P approximately NP? Deciding bipartite hypergraph coloring is NP-hard:


  While for bipartite graphs a 2-coloring can be found in linear time, it was shown by Lovasz [10] that the problem to decide whether a given k-uniform hypergraph is bipartite is NP-complete for all k≥3.


Bipartite hypergraphs are colorable in expected (average) polynomial time:


  The purpose of this note is to present an algorithm that colors a hyper-graph chosen uniformly at random from the family of all labeled, 3-uniform, bipartite hypergraphs on n vertices in O(n^5 * log (2n)) expected time. 


Does this imply that P is approximately NP? 

Source: https://www.math.uni-hamburg.de/home/schacht/abstracts/09eurocomb.pdf
No, it doesn't prove that P = NP.

This has nothing to do with approximations; it has to do with average-case hardness vs worst-case hardness.  The two results are showing that solving the problem for a randomly chosen hypergraph is usually easy; but there exist hypergraphs where the problem is hard.  Presumably, choosing a hypergraph at random is very unlikely to give you one that's hard, but they do exist.

NP-completeness is about worst-case hardness: even if a problem is easy on average, it can still be NP-complete.

I have no idea what "P is approximately NP" would mean in any precise sense.