
After almost 20 years, math problem falls - ph0rque
http://www.physorg.com/news/2011-07-years-math-problem-falls.html
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rabedik
"For an arbitrary polynomial function [...] determining whether it’s convex is
what’s called NP-hard. That means that the most powerful computers in the
world couldn’t provide an answer in a reasonable amount of time."

 _eye twitch_

A problem's complexity class only gives an indication of potential runtime.
There are decent algorithms for a lot of NP-complete problems (airline/package
routing, compiler backend optimizations, etc.) that don't require "the most
powerful computer in the world". I'm pretty sure the correct definition of
hardness/completeness in complexity theory can be boiled down to a sentence
that's much more correct than that, especially if you include something like
"at least as hard as all other NP problems"

~~~
ars
In math (well, in proofs) an answer must be "the" answer, a good enough answer
is not good enough.

Those algorithms you mention provide reasonable answers, but they do not
guarantee that the answer is the best answer.

~~~
mturmon
Your last statement is incorrect for one of the problems mentioned in the
parent comment ("airline/package routing" which I take to be linear
programming problems).

Simplex-type algorithms are typically used for these problems. These
algorithms also have exponential time complexity in worst-case. In practice,
well-designed simplex algorithms are of low-order-polynomial complexity. They
terminate with an exact optimum.

This case, and others, are examples of situations where conventional
complexity analysis fails to provide insight into real-world performance.

Last I knew, people were trying to analyze expected time complexity of certain
(analysis-friendly) simplex algorithms under distributional constraints on the
input structure matrices. But it's important to understand that this is
mathematical back-filling to explain what is already known to hold in
practice.

~~~
dododo
for linear programming there are several well known polynomial time algorithms
(some interior point methods: ellipsoid, projective, etc)--the problem itself
is by no means in EXP or NP.

for worst case complexities it's often just a small region of the problem
space that makes the bound large--but enumerating or cutting out those cases
is tricky. you just don't see them on average because they are few (c.f.,
quicksort: O(n^2) worst case, O(n log n) average).

~~~
mturmon
Yes, here's a summary (somewhat dated) of this, given in a prize award for one
of the key people who showed that worst-case is rare:

[http://www.informs.org/Recognize-Excellence/Award-
Recipients...](http://www.informs.org/Recognize-Excellence/Award-
Recipients/Karl-Heinz-Borgwardt)

(expand out the "show more" button). Of course, there has been more progress
since, but this summary is pretty good.

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stevetjoa
How pleasantly surprised I am to find that the work of Amir Ali Ahmadi, a
friend and former student at Maryland where I was his EE TA, appeared on
Hacker News. We both took the same graduate-level class on optimal control;
despite being an undergrad, he was one of the best students in the class.

The result is intriguing and reasonably accessible: for all (multivariate)
polynomials whose degrees are even and at least four, determining convexity
(of any type! strong, strict, regular, pseudo-, quasi-) is strongly NP-hard.
To prove this, the authors equate the problem of determining convexity to the
problem of determining the nonnegativity of biquadratic forms -- a known NP-
hard problem.

Find the paper here: <http://aaa.lids.mit.edu/publications>. (Edit: direct PDF
link
[http://mit.edu/~a_a_a/Public/Publications/convexity_nphard.p...](http://mit.edu/~a_a_a/Public/Publications/convexity_nphard.pdf))
See Table I on page 18 for a concise summary.

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gjm11
Like most things on physorg, this is blogspam. The original is at
<http://web.mit.edu/newsoffice/2011/convexity-0715.html> and (unlike the
physorg version) includes links to more detailed information about the
research.

~~~
podperson
It's also badly written, buries the lead, and exaggerates the solution (which
is that the math problem didn't fall, but a simpler problem which is almost as
useful did.

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seagaia
Another one added to the NP-hard list, mhm.

Maybe some things to clear stuff up:

NP-complete is a subset of NP-hard. NP-complete is entirely in NP, not all NP-
hard are in NP.

