Does Memcomputing really solve an NP-complete problem? I came across an article published in Science "Memcomputing NP-complete problems in polynomial time using polynomial resources and collective states", which makes some pretty astonishing claims. 


  Memcomputing is a novel non-Turing paradigm of computation that uses interacting memory cells (memprocessors for short) to store and process information on the same physical platform. It was recently proven mathematically that memcomputing machines have the same computational power of nondeterministic Turing machines. Therefore, they can solve NP-complete problems in polynomial time and, using the appropriate architecture, with resources that only grow polynomially with the input size. 


(Italics mine). 

I would dismiss this off the bat as non-serious, given the strong nature of the claims, if it weren't for the fact that this was published in Science, and that related material by some of the authors was published in Nature Physics, in an IEEE journal and in Physics Review E, all of which are reputable peer-reviewed publications that wouldn't let such claims get published without them being serious. 

So is it true? Can these people solve NP-complete problems in P-time using their model?
I feel this has been answered sufficiently in the comments, so to just sum everything up:


The authors do not claim P=NP, which is a statement about deterministic and nondeterministic Turing machines.
The authors propose a model of computation that they claim to show is equivalent in power to nondeterministic Turing machines.
The authors construct physical machines that implement this model of computation for small input sizes.
The authors argue that building larger versions is physically realizable/possible with polynomial-sized resources.
This last claim, which is of course not proven and not really a formal statement, would imply a that it is generally physically possible to solve NP-complete problems with polynomial-sized resources.
Scott Aaronson in a blog post explains why this last claim is problematic and why the scalability of their approach has problems: http://www.scottaaronson.com/blog/?p=2212