Tuesday, August 17, 2010

P ≠ NP (part 2)

As I mentioned in my last post, P and NP are commonly encountered complexity classes. Knowing what complexity class a problem is in can give us some idea of how difficult it may be to solve the problem. So what's the difference between P and NP? I don't want to get into the technical definition of P and NP, so I'm going to make broad generalizations that aren't technically rigorous, but will give you a feel for the difference.

Problems in class P are generally considered “tractable” by computer scientists. By this we mean that a computer can probably solve such a problem efficiently and relatively quickly. Recall that the complexity classes relate the ‘number of pieces’ to the difficulty. If you can solve a particular problem in P, then a similiar problem with a ‘larger number of pieces’ is likely within your grasp (and if not, it is usually a question of upgrading to a better computer). Problems in class P are the ‘meat and potatoes’ of computer programs.

Before I discuss NP, let me introduce the complexity class EXPTIME. EXPTIME is where you find some really hard problems, like computer chess or Go. It is true that there are computers that can play chess really well, but if we were to modify the rules of chess to add a couple of new pieces and make the board 9x9 instead of 8x8, it would greatly increase the difficulty of the game. Problems in EXPTIME are generally considered “intractable”. By this we mean that a computer will have a hard time solving such a problem efficiently and quickly. And if you are able to solve a particular problem in EXPTIME, then a similar problem with just ‘a few more pieces’ is likely beyond your grasp, no matter how fancy a computer you might buy. Problems in class EXPTIME take a lot of time and money to solve.

So what about NP? Problems in NP are quite curious. They seem to be very difficult to solve, much like EXPTIME problems, but they have the unusual property that it is very easy to check if a solution is correct. Let me illustrate: if I showed you a chess position and claimed that it was a good position for white, you'd have to do a lot of work to verify whether my claim was true. In fact, it would have to be about the same amount of work it took me to come up with the position in the first place. On the other hand, if I were to show you a jigsaw puzzle and claim that I had finished it, you could tell at a glance whether my claim were true. Problems in NP seem to be much harder to solve than problems in P, but as easy to verify as problems in P. That is a little weird.

Problems in NP are often encountered in computer programs, and many of these kind of problems, although very difficult to solve, have approximations that are relatively easy. In some cases, when a perfect solution is not needed, one that is ‘good enough’ will be a lot easier to compute. Another weird thing is that a lot of problems in NP don't sound at all like they'd be hard to solve. Here are some examples:
  • Take a group of people and divide them into two subgroups such that both subgroups have exactly the same amount of change in their pockets. (No, you aren't allowed to move the change around.)
  • Find the shortest path that visits all the major landmarks in a city.
  • Check if two computer programs always produce the same output.  Whoops!  I blew this one.  Check the next posting.
There is one final bit of weirdness. It is easy to prove that EXPTIME problems are much harder to solve that P problems, but no one has proven that NP problems are harder than P problems. (or that they aren't harder than P problems!) This isn't for lack of trying. Many smart people have worked on a proof for some time. There's a million dollar prize for the first person to prove this one way or the other. Recently, Vinay Deolalikar of HP Labs claimed that he has a proof. Unfortunately, other experts in the field have pointed out flaws in the proof that may invalidate it.
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