
Betting on whether P=NP  - michael_nielsen
http://rjlipton.wordpress.com/2009/09/15/theory-has-bet-on-pnp/
======
btilly
The thesis is well presented, but I don't buy it. Suppose that P=NP, but the
algorithm you find has running time O(n^100). How does the world change?

Obviously not much because nobody could actually run it. A theoretician would
respond that if Moore's law continued long enough, then eventually this would
be practical. However physicists tell us that there is a minimum amount of
energy released by irreversibly flipping a bit, from which we can calculate
the minimum amount of energy required to run this algorithm for an interesting
data set. That minimum makes this algorithm infeasible for the foreseeable
future, no matter what advances we make in technology.

While we say that we believe that P!=NP, the bet that we're making in real
life is that NP-complete problems are computationally infeasible.

(That said factoring has never been proven to be NP-complete, so we could come
up with a good factoring algorithm without being able to solve NP-complete
problems.)

------
amichail
How is trying to prove that P=NP distinctly different from trying to prove
that P!=NP?

Perhaps while trying to prove P=NP, you finally realize why it can't be the
case and end up proving that P!=NP instead?

~~~
brazzy
P=NP can be proven by example. All you need is one concrete algorithm. Proving
that no such algorithm can exist is a whole different beast and not likely to
be the result of trying to construct e.g. a factoring algorithm and failing.

~~~
amichail
_Proving that no such algorithm can exist is a whole different beast and not
likely to be the result of trying to construct e.g. a factoring algorithm and
failing._

Why is this obvious? Perhaps the search for an algorithm would give you a key
insight as to why P!=NP.

~~~
brazzy
Possible, but unlikely.

Look at Fermat's last theorem, a much simpler statement. All it would have
taken for a positive proof would have been four numbers. Trying to find them
by brute force failed and, really obviously, yielded absolutely no insights.

The actual negative proof took a brilliant mind, years of complete dedication
that might have turned out to be wasted, and the combination of two formerly
completely unrelated fields of mathematics.

~~~
roundsquare
Also, just take a look at some examples of lower bounds on a problem. Even the
simplest one I know (sorting):

<http://www.cs.colorado.edu/~karl/2270.spring03/sorting.html>

is much tougher than showing the running time of an algorithm.

