

"Holographic Algorithms" method threatens collapse of complexity classes (P could = NP!)  - injesus
http://pages.cs.wisc.edu/~jyc/papers/HA-survey.pdf
Very Interesting!
======
tlrobinson
For those who don't want to read the whole thing, here's the part I found most
interesting (though I didn't get through it all either...):

In this article, we will survey the new algorithm design method called
holographic algorithms. This method uses perfect matching as a basic coding
technique to encode computations, and then the FKT-algorithm to carry out the
final computation. A particularly innovative idea is to choose a set of linear
basis vectors to express and interpret a desired computation. In effect the
algorithm is designed to manipulate sums of perfect matchings in
superpositions, while the speed up is achieved by cancellations among such
"holographic mixes". These holographic algorithms are quite unlike anything
before, except perhaps quantum algorithms. At the heart of the computation is
a process of introducing and then canceling exponentially many computational
fragments. But unlike quantum algorithms, these holographic algorithms produce
classical polynomial time algorithms. So far this method has produced some
exotic algorithms for problems which were not known to be in P previously, and
minor variations of which are known to be NP-complete or NP-hard.

The most intriguing question is whether this new theory can lead to any
collapse of complexity classes. We contend that our belief of NP != P is based
on the sense and experience that the usual algorithmic paradigms are
insufficient for NP-hard problems (we don't have strong lower bounds for
general models of computation). But does our erstwhile experience apply to
these new exotic algorithms? If the answer is no, then it is conceivable that
the new methodology may lead to a radically revised conception of P vs. NP. Of
course it is quite possible that the theory of holographic algorithms does not
in the end lead to any collapse of complexity classes. But even in this
eventuality, as Valiant suggested in [54], "any proof of P != NP may need to
explain, and not only to imply, the unsolvability" of NP-hard problems using
this approach.

~~~
mxh
Two things leap out at me:

"So far this method has produced some exotic algorithms for ... minor
variations of [problems] which are known to be NP-complete or NP-hard."

Well, I guess all they have to show is that those "minor" variations don't
take the problems out of NP-complete or NP-hard .... that should be pretty
easy, right?

"'any proof of P != NP may need to explain, and not only to imply, the
unsolvability' of NP-hard problems using this approach."

Uhm - so, when and if someone gets around to the Turing-award winning business
of proving P != NP, that proof _may_ need to specifically address this
technique? And that makes the technique important?

At first blush, I think there's a snake around here somewhere that's a quart
low. The authors might be on to something, but they look bad in this extract.
They might have found something ("exotic algorithms for problems which were
not known to be in P previously") interesting, but P == NP seems like
overselling it, and that casts doubt on the value of the whole enterprise.

~~~
amichail
Welcome to theoretical computer science.

~~~
tlrobinson
Q: How many theoretical computer scientists does it take to change a light
bulb?

~~~
joshwa
A: <http://www.cl.cam.ac.uk/~tjr22/whiteboard-2007-07-18.jpg>

------
greendestiny
Take this with a grain of salt, because I'll admit I didn't read it very
thoroughly. My understanding of holographic algorithms is that they make a
primitive out of a matching algorithms that could be executed in constant
physical time with holographic storage. While thats great, I think it only
fair to include the complexity of the matching in the overall algorithms
complexity, because although it happens in constant time this only because of
massively parallel interactions in the storage.

Still I think when the author says that P != NP proofs will have to address
this I think he's right that that _is_ important. It means that this is at the
crux of the issue, either way it goes.

~~~
amichail
_My understanding of holographic algorithms is that they make a primitive out
of a matching algorithms that could be executed in constant physical time with
holographic storage._

Actually, from what I understand, that's not the point at all! What is
"holographic storage" btw?

~~~
greendestiny
<http://en.wikipedia.org/wiki/Holographic_data_storage>

The whole point being that the reference signal can be projected into the
storage and match found in a single pulse. Holographic algorithms deal
generally with algorithms based on interference, so they might not directly
relate to a physical storage device but the principles are similar.

~~~
amichail
The point is that this paper is about algorithms for classical (e.g., not
quantum or anything special) computers. That's why it is so interesting --
it's kind of like quantum algorithms but will work with standard computers.

