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Consequences of a Solution to NP Complete Problems?
m00nshyn3 asks: "If a person were to find a O(n) solution to an NP complete
problem, it would obviously be a great advance in computer science,
but what are the consequences of such a discovery? Would our most
popular implementations
of cryptography be useless overnight? It seems like there is a lot
of immediate damage that could occur if such a solution were found.
So if (when) the time comes, what is the responsible way for the
solution to be made public?" If you had such an algorithm in hand,
what could you do with it? It would be interesting to see how many
problems we could map into the NP Complete model.
Factoring is not NP_Complete. There were some early public key cryptsystems that were NP complete but they were abandoned rather quickly. While the best general case solutions to NP-Complete are O(2^n) there are many specific case solutions that will solve some subset of problems much faster. Or will get close to the correct result quickly. In a crypto-system you want to make sure that it is hard to solve ALL posible cases not just the worst case senario.
Erlang Developer and podcaster
Several router manufacturers are designing their new equipment with support for MPLS. MPLS, among other things, offers better support for traffic engineering. Traffic engineering means deciding which router hops to take to get from border point A to border point B (it also means you can select an alternate route depending on network congestion or backbone router failures).
Finding the most efficient path between the two endpoints is an NP complete problem.
I think this problem also exists for BGP/IGP networks, but I'm not experienced with them. I do know that the promise of the internet being secure and redundant and safe from one node being bombed is a bunch of nonsense. Although the technology to route around outages exists, the chore of houskeeping for it has turned out to be rather intractable so it is minimally implemented at best. Usually if a large backbone provider suffers a hardware failure and can't swap in a backup, they call in one of their router guru's to look at their topology map and configure their other routers to route data around the bad router at that time. That's the automated process.
All the providers out there would *REALLY* like a tool that could take their network topology map and output a reasonable set of LSPs (MPLS tags that indicate the route the packet is to take) for them in a reasonable amount of time. However, since this problem is NP complete, such a tool would have to compute every possible path and then choose those at the top of the list.
Education is a better safeguard of liberty than a standing army.
Edward Everett (1794 - 1865)
The simplest problem to understand in NP space is optimization. EVERY optimization problem is NP-complete. If you want to find the shortest route to visit every city in the US, this is an optimization problem.
What a problem being in NP-complete space means is that if you can solve one problem in the space, you can solve all problems in that space (by transforming the other problems to look like the solved problem and then solving that).
So if you prove P=NP, then you can solve optimization problems in polynomial time O(n^t). Pick a problem where you could say "I wonder what the fastest/shortest/lightest way of doing this is?" and you could solve it in O(n^t).
4 caveats/common misconceptions:
O(n^t) could be very VERY large. n^237203 still takes a very long time.
As far as I know, encryption algorithms are NOT KNOWN to be NP-complete. So it's UNKNOWN what proving P=NP means to the encryption world.
People who say P=NP is not provable aren't conveying the situation. The situation is that it is UNKNOWN if P=NP or not. They don't know the answer.
There are problems that do not fall into NP-complete class. There are NP problems that are not complete. There are (provably) problems that cannot be solved. So P=NP doesn't really act as a cycle multiplier (you don't just get more pure computation), it just gives you another trick in your bag.
So my answer for what happens if P=NP is proven: Everything in the world gets cheaper.
passetspike!