Theory of Computation: P and NP

I am currently studying the theory of computation, which is a fascinating subject. It turns out that there are some rigorous, mathematical truths behind computation that can tell us some extremely interesting things. For example,

  1. Can this problem be solved by computers?
  2. Can this problem be solved in  a reasonable time by computers?
  3. Can I turn this problem into another problem that can be solved (at all, or in reasonable time)?

Read on for a beginner’s explanation of P and NP.

P and NP were terms that I heard thrown around quite a lot by smarty-pants computer programmers, but they never meant much to me until now. Here is my understanding of P and NP. If I make any gross errors, please let me know in the comments. (Note: this explanation requires you to have a basic understanding of Big O notation. See here for a good explanation.) 

P is the set of problems that can be solved in polynomial time(ie O(n\^k) where k is some constant). NP is the set of problems that can be verified in polynomial time. Essentially, if a problem is in P, it is easy to solve (Theoretical computer science has a weird definition of easy). If it is in NP, then it is easy to check if a solution is correct or not, but we haven’t yet found an easy way to solve it. Our current solutions to problems in NP usually involve brute force exponential time algorithms with some heuristics thrown in to try to speed things up.

Showing something is in NP doesn’t mean it isn’t in P, but then you get into the whole P vs NP problem.

Six Degrees of Kevin Bacon is a problem in P: given a graph of people with edges representing relationships, is there a path of six or less edges that connects someone to Kevin Bacon? The polynomial time solution involves  a breadth-first search of the graph, stopping once Kevin Bacon has been reached or once all paths have six edges.

A good example of a problem in NP is finding the Hamiltonian path, which is the path through a graph that visits each node exactly once. It is easy to verify - just try it out and reject if any nodes were visited more than once or if any nodes were missed - but hard to solve.

Why do you care about this? You are working at a hot startup whose business plan is a mashup of the words social, Rails, local, crowd-sourced, and jacuzzi-full-of-benjamins. Obviously, it is a Facebook-killer, and as such, has a graph of profiles connected by friendships. Well, suppose your non-technical co-founder decides that the missing feature from your product is an iteration on the “Six Degrees of Kevin Bacon” idea: He wants to show the world that everyone on the site is connected together in one gigantic path that goes through each profile exactly once. “Easy!” you cry, as you look for the ActiveRecord method that does this for you. Then you remember that this is an example of the Hamiltonian path problem, which means that VC dollars will run out long before the path is found. Instead, you convince him that implementing Six Degrees of Kevin Bacon is a better idea. Now each user gets a sweet badge on their profile if they are within six degrees of Kevin Bacon, who is an avid user of your product.

P and NP are fundamental ideas in computer science. I’ll talk about NP completeness next, and then P (!)= NP. Let me know what you think in the comments.