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

[simont]

An "NP" problem is one that has no known set of steps for producing an answer, and thus requires you to check every possibility until you find one that's right.

Doing that is a set of steps for producing an answer! You probably meant to say "no efficient set of steps" here, or some such.

Also, technically, NP is a superset of P – there's no implication that "an NP problem" doesn't have a polynomial-time solution. Colloquial usage is to only describe problems as NP if they aren't (known to be) in P, on the pragmatic basis that if they were easier than that you'd have said so, but that's not what it really means.

I think the major thing I would add to this explanation is a paragraph explaining why the question is open, along the lines of:

At first sight it seems obvious that NP is a bigger class of problems than P, or in other words, that when a problem looks as if it needs you to go through all the possibilities one by one it's likely that there is in fact no quicker way to solve it. However, it turns out that this is surprisingly hard to prove (or disprove), so there's still a possibility in principle that the classes P and NP might turn out to be identical – which would mean that any problem for which you could efficiently check a candidate solution would also have an efficient way of finding a solution from scratch. Most computer scientists believe this is not true, and that P and NP are just as different as they appear, but a proof would be big news either way. (Though it would be bigger news if it went the surprising way and said that P and NP were equal.)

[andrewducker]

Cheers - I've updated it, does it make more sense now?

[simont]

It still says that a non-P problem is one for which there's no known efficient algorithm. The whole point of P possibly being equal to NP is that all those problems might turn out not to be non-P after all :-)

Perhaps move the bit about our limited knowledge down to the TSP sentence? So that a non-P problem is simply described as "one which has no efficient set of steps", and then "A common example of a problem which is thought to be hard in this way is The Travelling Salesman problem...".

[andrewducker]

Cheers - I'll make those changes when I get back from a one-to-one with my manager :->