(This post is similar to
this old post.
I am posting this anyway since when I first posted I made fundamental mistake. I fixed it
the point I was trying to make get lost.)
About once a semester I get asked
What are the natural problems that are in NP,
not known to be in P, and not known to be NPC?
What is known about them?
And I give the standard answers:
Graph Isomorphism. If its NPC then PH collapses
so it is though to NOT be NPC. There is no real consensus about
its status with respect to P.

Factoring. If its NPC then NP=coNP so it is thought to NOT be NPC.
Most people think that it is NOT in P since people really want to solve
this one and have not been able to. This is not a rigorous argument.
Factoring is in QP.
If quantum computers are ever practical then we may need to rethink
how we think about these things.

Discrete Log. Actually, are there reasons to think this is NOT NPC?
The answer above are standard and I suspect most of my readers know them.
However, there is a large source of problems that are in NP, likely not NPC,
likely not in P, that people don't seem to talk about much: SPARSE PROBLEMS
that are thought to be hard. The ones I know about are from Ramsey Theory.
I give one example but there are many like it:
{(1^{n},1^{m},c) : the n×m grid can be ccolored and not have any mono squares}

This is clearly in NP since the coloring itself is the witness.

Since this is a sparse set it is not NPC unless P=NP.
(Actually it can be coded as a tally set.)

I want to say People think this is hard. You may ask
Name Three of Them! Indeed, the number of people who work on Computational Ramsey Theory
is fairly small. However, Ramsey Theory itself has many people working on it and
nobody has anything close to a result indicating that this problem is in P.
I personally think that it is hard.

For a fixed c there is a finite number of grids G1,...,Gp
such that the n×m grid is ccolorable iff none of G1,...,Gp can fit inside the n×m grid.
Hence, for fixed c, the problem is in P.
(So the problem is Fixed Parameter Tractable.)

Everything said above holds if you replace squares with rectangles or other configurations in the above.
Questions:
Are there other sparse problems in NP that we think are NOT in P?
(that is, ones that DO NOT come from Ramsey Theory).

Should we be teaching these in our complexity classes as examples of possible
intermediary problems? The proof that they are likely not NPC is easy.
However, to argue that these problems are likely not in P is hard.
Also, these problems may appear unnatural to some students.
(They may appear unnatural to some of my readers.)

It is easy to argue that Factoring is probably not in P since you can point
to the fact that people REALLY want to solve it quickly and so far have not.
This is not a rigorous argument but I do count it as evidence.
Is there a similar argument for GI?
Is there a similar argument for my Ramsey Problems?

Is there a mathematical reason to think that Factoring, GI, or my Ramsey Problems
are not in P?

Posted By GASARCH to Computational Complexity at 7/15/2010 10:20:00 AM