
Stating P=NP Without Turing Machines  - wglb
http://rjlipton.wordpress.com/2010/06/26/stating-pnp-without-turing-machines/
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faragon
Can you wait for a month?

<http://news.ycombinator.org/item?id=1257488>

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tome
Does anyone know what the unsolved problems that Dantzig solved actually were?

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adamo
[http://en.wikipedia.org/wiki/George_Dantzig#Mathematical_sta...](http://en.wikipedia.org/wiki/George_Dantzig#Mathematical_statistics)

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tome
Yes I found that myself, but it doesn't actually state what the problems were!

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tkahn6
This is a great explanation. I found it really easy to follow.

The only part I'm having trouble understanding is the cubic graph example. I
am aware of the four-color map theorem but I've never seen it expressed
mathematically.

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tkahn6
Also, even if we proved P=NP wouldn't we still have to discover a general
algorithm for solving IP systems?

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eru
We do have general algorithms for solving Linear Integer Programs. They just
take some time, because IP is in NP. (It's also in Co-NP, if I remember
correctly.)

If P=NP, and somebody has shown a polynomial algorithm for any one problem in
NP [1], then we can construct a polynomial algorithm for IP.

Of course, getting a good polynomial algorithm for IP would still be an
interesting problem. E.g. bubble sort is polynomial, but far from the ideal
sorting algorithm for most applications.

We also have lots of algorithms to solve IP already, that work well in
practise for many types of problems. But all of them become slower than
polynomial for certain pathological inputs.

Interestingly, no polynomial simplex algorithm (for continuous linear
programming) is known, but in practise simplex often performs faster than most
polynomial algorithms for solving LP.

