

MIT Engineers Beat the Complexity of Deep Parallel Processing - futuristdata
http://motherboard.vice.com/read/mit-engineers-offer-a-new-solution-to-one-of-computings-most-complex-problems

======
graycat
The article has:

"The problem is NP-hard, actually, which is a designation applied to only the
most difficult and unsolvable computing problems."

Nonsense. NP-hard problems are not "unsolvable". Instead, for any NP-hard
problem, e.g., the traveling salesman problem, it's easy to write software
that will obtain an optimal solution to any instance of the problem in finite
time and often in practice quite reasonably short time. Nearly anyone who goes
on a shopping trip to the grocery store, drug store, and lunch at a fast food
restaurant gets an optimal solution just by hand.

In fact, clearly, easily, we get exactly optimal solutions to NP-hard
optimization problems routinely millions of times a day, just by hand, even
more easily and frequently by computer.

~~~
markpundmann
Your example has 3 locations in it. Of course it isn't hard! Try doing that
for a billion locations.

~~~
graycat
I just responded to what the OP stated, in particular the claim of
"unsolvable". The OP said nothing about 1 billion locations.

If the billion locations are all in a straight line, then not a problem!

For the set of real numbers R and a positive integer n and R^n with the usual
inner product, norm, and topology, if the billion points are in R^n then there
is a super cute algorithm: Find the minimum spanning tree (with a _good_
algorithm) of the billion locations and then, for the solution, just traverse
the tree never returning to a location except at the end the first. Then for
any x > 0, no matter how small, as the number of locations grows, and a
billion is quite high, the algorithm will find a solution within x% of
optimality with probability greater than 1 - x.

What's _hard_ about NP-hard, e.g., for the traveling salesman problem on m
locations, is that no one knows how to write software that will solve, with an
exactly optimal solution, worst case problems in time proportional to a
polynomial in m. But that's nothing like what the OP claimed.

I was just responding to the OP.

Yes, 0-1 integer linear programming is also NP-hard. The last case I attacked
had 600,000 variables and 40,016 constraints, and I wrote some software using
Lagrangian relaxation, e.g., as in

[https://news.ycombinator.com/item?id=8919311](https://news.ycombinator.com/item?id=8919311)

and got a feasible solution within 0.025% of optimality in 905 seconds on a 90
MHz PC.

Lesson: NP-hard does not always mean hopeless or "unsolvable" in practice.

Besides, for 1 billion locations, that would be one heck of a long time away
from home for the salesman! Besides, as in nearly all real world optimization
problems, the goal is to save resources, e.g., time, money, and saving all but
the last penny is fine. Or, why spend billions to save the last penny?

Actually this point is important: Early in Garey and Johnson is a cartoon
where a mathematician is reporting to an executive, presumably one at Bell who
wants to know the least cost way to build a communications network, that he,
the mathematician, cannot _solve_ the executive's problem but neither can a
long line of other mathematicians.

Nonsense. All the executive wanted to do was design a network and save money,
and that goal was quite likely quite doable.

Instead, sorry, it was as if the mathematician was looking for a long term
job, that is, claiming that the executive should fund the mathematician to get
a _solution_ so far not possible. In a word, the mathematician was lying in
order to land a cushy job, for life!

For nearly anything, and not just NP-hard problems, say, a billion, too much
of it is _hard_.

Yes, the question of P versus NP is one of the most astounding unsolved
problems there is. The darned thing seems to be of even philosophical
significance comparable with the work in set theory of Kurt Gödel and Paul
Cohen, etc.

And, yes, Clay Math in Boston has a $1 million prize!

Yes, Scott Aaronson has much more to say!

Ah, somewhere I still have Max Zorn's (as in Zorn's lemma version of the axiom
of choice) copy of Cohen's paper! I got it from Zorn when at a tea I asked him
what Cohen had proved! So, yes, I have some interest in something of such
philosophical interest!

Still, the claim in the OP that NP-hard problems are "unsolvable" is nonsense
or needs a nonsense definition of _solvable_!

