A simple explanation to P and NP

[andrewducker]

lebeautemps asked for a simple explanation of the whole P=NP thing that's been circulating the internet recently. I figured I'd give it a go - I'd appreciate it if the more knowledgable on my friends list could correct anything particularly stupid I'd said...

A "P" problem is one that is easily solved through a series of steps, without trying every single combination - such as "sort a bunch of names into alphabetical order". You don't have to try every single possible list of names in order to sort a list of names, you can just use a series of comparisons to shuffle them up and down until they're sorted.

An non-P problem is one that has no known efficient set of steps for producing an answer, and thus requires you to try possibilities until you find one that's right. A common example of one that is thought to be hard is The Travelling Salesman problem - "given a bunch of cities and a bunch of roads connecting them, what is the shortest route that will take the salesman to each city exactly once?" There's no known solution to this problem other than "start trying solutions and keep going until you've found one." (although there are ways of excluding obviously wrong answers quickly).

NP is the superset of all problems that are easy to check, P is the smaller subset of all problems that are easy to solve, and proving that they are the same thing would mean that all problems that are easy to check are also easy to solve.

One of the reasons this is important is that pretty much all of the security methods we use online rely on things like "integer factorization" not being P - the fact that we can multiply two large numbers together to get an answer very quickly, but breaking that large number back into the two component parts requires every possibility to be checked (which takes years/decades for very large numbers).

The recent fuss is because of a paper that was published claiming that P!=NP - i.e. that there are definitely problems that are easy to check, but not easy to solve. This would mean, for example, that there is no simple way to convert the big number back into its two components, and thus that all of our security is safe (from that particular direction, anyway).

Comments

[ciphergoth.livejournal.com]

I would drop all talk of trying lots of solutions; it's not just wrong in the details, it gives the wrong impression. For example, trying lots of solutions has nothing at all to do with how we actually attack integer factorization. Also you assume that all problems are NP - some problems, you can't even check the solution.

A P problem is one that can be solved in poly time.

An NP problem is one where you can check the solution in poly time.

If P != NP, there are problems that don't have efficient solutions, even though if we saw the solution we could efficiently check it.

[andrewducker]

I don't assume that all problems are NP - I say "NP is the superset of all problems that are easy to check".

I'm not going to mention poly time - the whole point was not to turn off people who go blank at the mention of things like that. I need to use examples that are in simple english, so that people can grasp at them. Without examples most people don't grasp things.