Second Clay Mathematics Award claimed, but how should it be split?

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From: Future_News on 21 Oct 2009 12:34

---

The world of theoretical computer science has been turned upside down
by a stunning triple development which has finally solved its most
famous problem: whether P=NP. This was one of the seven problems for
which the Clay Mathematics Institute offered a million dollars in
their famous Millennium Meeting over thirty years ago, and the second
(after the Poincaré conjecture) to be solved. However, a controversy
has erupted over how the prize money (which ten years ago was
increased to two million dollars) should be distributed.

What does the statement `P=NP' mean? A good way to understand it is to
look at an illustrative example. Suppose I secretly choose two prime
numbers p and q, tell you their product n=pq and ask you to tell me
what p and q are. In principle you can work this out, since by the
fundamental theorem of arithmetic p and q are uniquely determined.
However, in practice it is not easy to come up with a systematic way
of finding them other than to look at all primes up to n1/2 and see
whether they are factors of n. If n has 500 digits, this takes much
too long to be feasible on even the fastest computers, and this
difficulty is the basis of much of modern cryptography. Actually, more
efficient methods have been discovered than straightforward trial and
error, but, even after decades of intensive research, it is still out
of the question to factorize a 500-digit number. At the end of the
last century there was a great deal of excitement when Peter Shor
discovered that quantum computers could factorize large numbers far
more efficiently, but, despite several well-publicized announcements
to the contrary, nobody has managed to build such a computer.

However one looks at it then, factorizing large numbers is not easy.
However, the following related problem is much much easier: I choose
p, q and n and ask you whether n=pq. All you have to do to answer this
problem is a boring long multiplication, and if you are careful, or
use a computer, you can do this quickly and accurately. Thus, although
it looks hard to find p and q, it is much easier to check whether pq=n
once p and q have been found.

There are many other problems in computer science of a similar nature.
They ask you to find an object with certain properties, and while this
looks extremely hard, it is easy to check whether a given object has
the required properties. One interesting example is the following: I
give you a mathematical statement S and ask you to find a proof of S
of under a certain length (written in a formal language of your
choice). One of the great fascinations of mathematics is that it is
(in principle anyway) far easier to check whether a proof is correct
than it is to find the proof in the first place.

The statement `P=NP' is roughly speaking the extraordinary claim that
when a property is easy to check, it is also easy to determine whether
any object has the property. Thus, if P=NP, then from the fact that
long multiplication is easy it would follow that factorizing large
numbers was easy, and from the fact that proofs are easy to check it
would follow that they are easy to find. Despite the strongly
counterintuitive nature of these conclusions, they are now known to be
true - but with qualifications which will be described below.

As mentioned above, this breakthrough came in three stages. First, in
a little known paper of 2019, Yuri Kolpakov, a Russian topologist,
found a higher-dimensional generalization of Frank's celebrated knot-
untying algorithm of three years earlier. Following Frank's general
scheme, Kolpakov showed that, given a hypersurface in Rn, one could
associate with it a graph with edge-weights belonging to a certain
ring. It was then possible to prove that the hypersurface was knotted
if and only if the graph had a property which, by a generalization of
classical results of Robertson and Seymour, was equivalent to the non-
existence of certain weighted subgraphs known as forbidden proper
minors. Testing for these can be done in polynomial time.

The technical details need not concern us here, but one point is worth
noting. The proof of Robertson and Seymour does not actually provide
the forbidden minors: it merely proves indirectly that they exist. The
same applies to the generalization. Consequently, Kolpakov's proof,
like Frank's before him, does not actually give any bound on the
degree of the polynomial which itself bounds the time taken by his
algorithm.

The second step was made two years ago by Jane Nichols, a graph
theorist at Yale. She was studying the computational complexity of
certain problems in graph theory, and while doing so discovered a
complicated property of graphs, again with weights in certain rings,
which she called, ironically in the light of later developments,
`pseudo-twistedness'. She showed that this property was NP-complete.
This, in brief, means that if an efficient algorithm could be found
for discovering whether a graph was pseudo-twisted, then P would equal
NP. Like almost everybody else, Nichols took this as convincing
evidence that detecting pseudo-twistedness was intractable, and moved
on to other research. Her paper also received little attention, and it
is interesting to speculate why this was. It has been suggested that
the phrase `pseudo-twisted' was somehow off-putting to most
mathematicians, causing them to believe that Nichols's research was
not well motivated, though in a recent paper, she has explained the
very interesting line of thought, originating in theoretical physics,
that led her to her definition.

Nichols gave several talks on her work, including a colloquium at MIT,
which was attended by Mike Stearns, a young topologist who had been
studying the work of Kolpakov. As he put it later:

As the talk progressed, an idly outrageous thought occurred to me
- wouldn't it be amusing if Kolpakov's untwisting algorithm could
detect Nichols pseudo-twistedness of graphs? At first I chuckled
inwardly - but then I began to feel the hairs on the back of my neck
standing on end. I skipped the tea after the talk and rushed back to
my apartment, where I stayed until I more or less collapsed into bed
at 4am. I got up the next morning feeling totally sure that there must
be a mistake, but everything seemed to check out.

What Stearns had done was to find a `polynomial reduction' from
Nichols's problem to Kolpakov's. That is, in effect he devised an
efficient algorithm for transforming a graph into a high-dimensional
analogue of a knot diagram, in such a way that the graph was pseudo-
twisted if and only if the `knot diagram' was genuinely knotted.
Kolpakov's powerful algorithm could thus be used to determine whether
graphs were pseudo-twisted, even though this had been shown by Nichols
to be an NP-complete problem.

Stearns's initial reaction was to go out and celebrate, but soon
things were to turn sour.

At first I was disbelieving. Then, after a few days of careful
checking, I convinced myself that the argument was sound. I then wrote
it up quickly and asked one or two colleagues to check the details
themselves. Unfortunately, word got out about what was going on, and I
suddenly found myself at the receiving end of a whirlwind of
attention, most of it unwelcome. People started to say that my
contribution to the problem was trivial: essentially nothing more than
putting two pre-existing pieces of work together. But if that was all
it took, then why didn't Nichols solve the problem two years ago?

Here is Nichols's view of the matter.

Yes, Stearns deserves credit for noticing that Kolpakov and I had,
between us, proved that P=NP, but no, his reduction is not a difficult
piece of mathematics. The way I like to look at it, the proof that
P=NP comes in two parts, Kolpakov's polynomial-time algorithm and my
proof of NP-completeness. It is not quite trivial that these apply to
equivalent problems, and I fully admit that I did not recognise the
importance of what I had done. Nevertheless, I think Stearns's
achievement was one that would have been made by whoever was the first
person to think of trying it. Certainly, as soon as I heard what he
had done, I was able to come up with a reduction myself without seeing
his preprint.

Don Jackson, one of Stearns's colleagues at MIT, hotly disputes this
version of events.

Nichols is understandably bitter that she was not the person who
solved the P=NP problem. However, the fact remains that if Stearns had
not existed, then it would almost certainly still be unsolved today.
Sometimes in mathematics, it is not the technical achievement that
matters, but the courage and vision to ask the big questions.

What of Kolpakov? He has been keeping his own counsel so far, but one
can well imagine his reaction at hearing that he is likely to share a
prize of two million dollars. The decision about how to split the
money is made by the scientific committee of the Clay Mathematics
Institute. When we contacted a representative yesterday, she said:

The prize will not be awarded until the result is published in a
reputable journal, and a committee, which is yet to be formed, has
deemed it to be correct. This process will take at least two years,
and until then the Institute has no comment about how the prize will
be shared.

Leaving aside the question of the exact apportioning of credit, and
surely all three mathematicians deserve our huge admiration for this
remarkable achievement, what is the significance for mathematics and
computer science? Here again there is a wide divergence of opinion.
Jim Davies, a computer scientist at Stanford, believes that the
importance of the P=NP problem has been exaggerated.

So there's a polynomial time algorithm for factorizing integers,
proving theorems, finding Hamilton cycles etc. And then what? The
degree is not just large, it is unknown. Probably if it was known then
it would be enormous. Even an n10 algorithm is useless for practical
purposes. So nobody's going to be put out of business just yet.

Don Jackson disagrees.

This is the beginning of the end. Even if the algorithm isn't
practical, it delivers a shattering psychological blow to the
mathematical community. We may still be better than computers at doing
mathematics, but we are not that much better . In any case, now that
the existence of an algorithm is known, the hunt is on for an explicit
example. Historical precedent and the enormous commercial implications
of being able to solve NP-complete problems suggest that this story is
set to run and run.

Only time will tell. Meanwhile, if you are struggling to prove a
theorem, you'd better hurry up if you want to get there before some
clever computer programmer does it first.

From: OwlHoot on 21 Oct 2009 16:05

---

On Oct 21, 5:34 pm, Future_News <future_n...(a)brew-master.com> wrote:
>
> [..]

Nice wind up ;-)

I fell for it hook line and sinker, even speed reading that year
as 2009. The author's posting nick should have been a giveaway too!

Mind you, I very much doubt things will turn out like that: P = NP
and similar problems have "undecidable" painted all over them IMHO.

Failing that, someone will prove that P and NP problems are disjoint
but there are types arbitrary close to each other, with a suitable
measure, so that there isn't a sharp dividing line (although in
that case the outcome would still be P <> NP fair and square).

But don't take my opinion as gospel, or anything like, as I'm fairly
clueless about all this!


Cheers

John Ramsden

From: Gordon Stangler on 21 Oct 2009 16:22

---

On Oct 21, 3:04 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> OwlHoot <ravensd...(a)googlemail.com> writes:
> > Mind you, I very much doubt things will turn out like that: P = NP and
> > similar problems have "undecidable" painted all over them IMHO.
>
> Why is that?
>
> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon mann nicht sprechen kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Yes, why must they be undecidable? This is a very deep problem, and
like most deep problems, I feel in my gut that they MUST have a
solution. I am of the opinion that P != NP. There are simply too
many classes of exotic problems for them all to have one optimal class
of solutions. That is, they are not all solvable in polynomial time,
no matter what they polynomial exponent is.

From: Future_News on 21 Oct 2009 18:42

---

On Oct 21, 3:11 pm, James Waldby <n...(a)no.no> wrote:
> On Wed, 21 Oct 2009 09:34:50 -0700, Future_News wrote:
>
> ...> The second step was made two years ago by Jane Nichols, a graph theorist
> > at Yale. She was studying the computational complexity of certain
> > problems in graph theory, and while doing so discovered a complicated
> > property of graphs, again with weights in certain rings, which she
> > called, ironically in the light of later developments,
> > `pseudo-twistedness'. She showed that this property was NP-complete.
>
> ...
>
> There's a slight problem regarding facts in the above paragraph.  Jane
> Nichols is *not* a graph theorist at Yale, she teaches Clown there.
> Ref. <http://www.clownlink.com/2007/10/jane-nichols.html>.  This is
> not to take away from the achievement, merely to set the facts straight.
>
> > Nichols gave several talks on her work, including a colloquium at MIT,
> > which was attended by Mike Stearns, a young topologist who had been
> > studying the work of Kolpakov.
>
> Some recent work of Stearns re "symbolic structures" and "symbolic
> overlays" is referenced at <http://www.michaelstearns.com/>.
>
> ...>     Nichols is understandably bitter that she was not the person who
> > solved the P=NP problem. However, the fact remains that if Stearns had
> > not existed, then it would almost certainly still be unsolved today.
> > Sometimes in mathematics, it is not the technical achievement that
> > matters, but the courage and vision to ask the big questions.
>
> ...
>
> --
> jiw
> :)

Thank you for your good spirited replies. I think the solution must be
perhaps two fold playing on the duality of function between software
and hardware.

Here is a loose schematic of this 'future' news idea: and the
'twisted' nature of it:

Architectural partition for a 100 GbE link
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Physical Coding Sublayer (PCS)
Physical Medium Attachment (PMA)
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sagesaqmuAimesnacwek0iectrahinntetrblh.eeedWcueocixmtephspo!
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(hh1eeer)
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(apotatitetsrr)
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artegpsimm,atalaiorlsyiakrf,ymeatbhetwyphterehaeenraddsebtjeuuaeatrssresatterpilmnep
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(lote|uoaaindtvtrgnhwieatoi,eRhnthmnr1riTieoot9cefRnhuhomisegr0Ltaoh)
h;amrmrertdylRieciooeemmeomuqusnbgeunootsihatr
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trdeaMahevncissheetonnieionleturmvct.ttrehiiTedooeoannhsburseiynnfyidocgnaerecrdttgmtrheaaaeepoamixntrsihntmeiony,pnagddur-
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From: Tegiri Nenashi on 21 Oct 2009 19:32

---

On Oct 21, 8:34 am, Future_News <future_n...(a)brew-master.com> wrote:
> What does the statement `P=NP' mean? A good way to understand it is to
> look at an illustrative example. Suppose I secretly choose two prime
> numbers p and q, tell you their product n=pq and ask you to tell me
> what p and q are. In principle you can work this out, since by the
> fundamental theorem of arithmetic p and q are uniquely determined.
> However, in practice it is not easy to come up with a systematic way
> of finding them other than to look at all primes up to n1/2 and see
> whether they are factors of n. If n has 500 digits, this takes much
> too long to be feasible on even the fastest computers, and this
> difficulty is the basis of much of modern cryptography. Actually, more
> efficient methods have been discovered than straightforward trial and
> error, but, even after decades of intensive research, it is still out
> of the question to factorize a 500-digit number. At the end of the
> last century there was a great deal of excitement when Peter Shor
> discovered that quantum computers could factorize large numbers far
> more efficiently, but, despite several well-publicized announcements
> to the contrary, nobody has managed to build such a computer.

I have an issue with the number encoding schema. Somewhere in the
introduction section Gary-Johnston textbook describes how numbers are
encoded in the computer and asserts that exponential notation
(regardless of base) is the most natural way to go. Objection, your
honor! There is no genuine mathematical reason why to exclude unary
number system.

Factorization problem is not hard in unary system. Any other NP-
complete problem that doesn't depend on number encoding? Let's try the
subset sum problem. On the first sight, the subset property looks hard
to check by the virtue of the fact that we have to analyze the
powerset which size is exponential (in any number encoding schema).
Let's look into the problem little closer. Consider the following
naive algorithm where we keep track of all the sums, and when adding a
number to the list we just go through the list and add the number to
each element. The list would double at each step,- you may say. Not
really. In unary number system there is just not enough room: one has
to have big numbers in order to accommodate all possibilities. So the
subset sum problem appears to be of cubic complexity in unary number
system.

So my question is: are there any genuinely "hard" problems that don't
depend on (rather arbitrary) choice of number system?

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