
Computer Scientists Find New Shortcuts for Traveling Salesman Problem - co_pl_te
https://simonsfoundation.org/features/science-news/computer-scientists-take-road-less-traveled/
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wolfgke
The general TSP (where negative edge weights are allowed) cannot be
efficiently approximated (if not P = NP).

The Christofides algorithm considers are far more limited version of the TSP
problem (whose decision problem is still NP-complete): the so called "metric
TSP": here the edge weights form a metric on the arcs, i. e.

d(v, w) > 0 and d(u, v) + d(v, w) >= d(u, w) for all u, v, w pairwise
different (where d(v, w) is the weight on the edge {v, w}).

~~~
bhickey
If you restrict the metric case even further and only allow planar graphs, you
get even some neat approximations. For example, there exists an O(c^(1/e^2)n)
epsilon approximation scheme:
<http://cs.brown.edu/~pnk/publications/tsp2005.pdf>

This manuscript, <http://planarity.org/> , contains most of the material in a
planar graph optimization course I took a few years ago.

------
maeon3
And by the way, if you do discover this "short cut" for the TSP. AKA you can
solve TSP for every case in polynomial time, your name will go down in history
for the following:

1\. Breaking every single encryption algorithm and brute-force security
methodology.

2\. Creating a strategy for completing ALL NP-Hard problems in polynomial
time, millions.

3\. Thousands of equations with rewards for solving them will suddenly become
solvable with your smart phone. You can collect on billions in reward money.

4\. You will have likely shattered some fundamental assumptions we have about
what matter and energy is in this universe.

So if you did solve TSP in polynomial time, you would immediately be the most
famous person in the computing world for a very very long time. I think there
is a solution to P=NP, but you are going to have to toy around with the fabric
of space-time itself to do so. Your algorithm will have to allocate quantities
of spacetime for itself to process in, and transport the answer back through
time to us. This will no doubt cause problems in subspace, but that's not our
problem.. yet.

~~~
martinced
_"Breaking every single encryption algorithm"_

Hardly.

For a start one-time pad (say using XOR) is a perfectly valid encryption
algorithm (and its actually used in the wild) and won't be affected a iota by
P = NP. Same for Carter-Wegman.

Then:

 _"If P=NP, you could still have a cipher that decrypts in linear time with
the key and n^1000 time without the key. So it's breakable in polynomial time,
yet cryptographically secure."_

Taken from Schneier:

<http://www.schneier.com/blog/archives/2010/08/p_np_1.html>

I don't disagree with the fact that it would be a major discovery but I very
much dispute your overgeneralization about the implication regarding
cryptography : )

~~~
maeon3
Security changes that will be required if P=NP is proven:
[http://security.stackexchange.com/questions/12802/how-
will-s...](http://security.stackexchange.com/questions/12802/how-will-
security-need-to-be-changed-if-p-np)

