[urgoroger]

That's correct. Given a polynomial time approximate algorithm for Ek-SAT (note, by approximate algorithm I mean along the lines of the formal definition of approximate algorithm, that the solution given by the algorithm falls in some fraction of the real answer for all instances, see https://en.wikipedia.org/wiki/Hardness_of_approximation), you would show P=NP.

My first concerns, namely the usage of analog methods to solve NP-complete problems, lies along the same lines as this post: https://www.scottaaronson.com/blog/?p=2212

Moreover, from what I can see, and as you mention as your original concern, the extent of proof for the 'exponential-speedup' claim exists in the form of some benchmarks, hardly a proof that their method actually works on all instances, which would be needed to show that it is a approximation algorithm as commonly defined.

The purpose of my post was to highlight that for some problems (for example, TSP), even a really crappy approximation algorithm would imply P=NP. As highlighted by the authors of this paper, MAX-EkSAT is in a similar situation, though judging by https://www.cse.buffalo.edu/~hungngo/classes/2006/594/notes/..., unlike TSP, approximation to SOME ratio is possible, though there exists aratio >1 which cannot be beat unless P=NP.

I was simply trying to address the statement: "So it appears that they're finding approximate solutions, so already it's rather less significant than the title suggests. ", since an actual approximation algorithm for some ratio really WOULD be significant (showing P=NP).

[joshuamorton]

I don't think your first statement is true, there exist polynomial time approximation schemes and approximation algorithms for np complete problems, notably a PTAS for knapsack. In other words, an approximation of one np complete problems doesn't imply an approximation of all of them.

I can't fully articulate the reasons for this though.

[zmonx]

The reason for this apparent discrepancy is found in the difference between strong and weak NP-completeness.

The fully polynomial-time approximation scheme (FPTAS) for the knapsack problem only runs in so-called pseudo-polynomial time:

https://en.wikipedia.org/wiki/Pseudo-polynomial_time

This means that the runtime is polynomial in the numeric value of the knapsack. Since the encoding of that numeric value only takes logarithmic space (unless you are using unary encoding), the runtime is in fact again exponential in the size of the input.

For this reason, the knapsack problem is called weakly NP-complete:

https://en.wikipedia.org/wiki/Weak_NP-completeness

One can show that, unless P=NP, a so-called strongly NP-hard optimization problem with polynomially bounded objective function cannot have a fully polynomial-time approximation scheme:

https://en.wikipedia.org/wiki/Polynomial-time_approximation_...

SAT, Hamiltonian circuit etc. are strongly NP-complete:

https://en.wikipedia.org/wiki/Strong_NP-completeness

Thus, an FPTAS for these problems would indeed imply P=NP.