[disclaimer: not an expert in any of this]
The ACM STOC 2018 conference links to "The Adaptive Complexity of Maximizing a Submodular Function" http://dl.acm.org/authorize?N651970
A DOI URI would be great, thanks.
There's no claim to have changed the complexity class, so"exponentially faster" just means the problem is still exponentially in problem size, but the exponent is smaller.
From the looks of it, though, it is a nice result because it has pretty wide applicability.
I'm not sure, but I doubt this algorithm is "faster" than the previous one on a single processor.
(The article is drivel; I couldn't figure out what they were talking about either, until I looked at the paper)