Theory of Computation

Hoooo, doctoral comp flashback. Okay, here goes.

We start with the idea of a decision problem, a problem for which an algorithm can always answer "yes" or "no." We also need the idea of two models of computer (Turing machine, really): deterministic and non-deterministic. A deterministic computer is the regular computer we always thinking of; a non-deterministic computer is one that is just like we're used to except that is has unlimited parallelism, so that any time you come to a branch, you spawn a new "process" and examine both sides. Like Yogi Berra said, when you come to a fork in the road, you should take it.

A decision problem is in P if there is a known polynomial-time algorithm to get that answer. A decision problem is in NP if there is a known polynomial-time algorithm for a non-deterministic machine to get the answer.

Problems known to be in P are trivially in NP --- the nondeterministic machine just never troubles itself to fork another process, and acts just like a deterministic one. There are problems that are known to be neither in P nor NP; a simple example is to enumerate all the bit vectors of length n. No matter what, that takes 2ⁿ steps.

(Strictly, a decision problem is in NP if a nodeterministic machine can arrive at an answer in poly-time, and a deterministic machine can verify that the solution is correct in poly time.)

But there are some problems which are known to be in NP for which no poly-time deterministic algorithm is known; in other words, we know they're in NP, but don't know if they're in P. The traditional example is the decision-problem version of the Traveling Salesman Problem (decision-TSP): given the cities and distances, is there a route that covers all the cities, returning to the starting point, in less than x distance? It's easy in a nondeterministic machine, because every time the nondeterministic traveling salesman comes to a fork in the road, he takes it: his clones head on to the next city they haven't visited, and at the end they compare notes and see if any of the clones took less than x distance.

(Then, the exponentially many clones get to fight it out for which ones must be killed.)

It's not known whether decision-TSP is in P: there's no known poly-time solution, but there's no proof such a solution doesn't exist.

Now, one more concept: given decision problems P and Q, if an algorithm can transform a solution for P into a solution for Q in polynomial time, it's said that Q is poly-time reducible (or just reducible) to P.

A problem is NP-complete if you can prove that (1) it's in NP, and (2) show that it's poly-time reducible to a problem already known to be NP-complete. (The hard part of that was provie the first example of an NP-complete problem: that was done by Steve Cook in Cook's Theorem.)

So really, what it says is that if anyone ever finds a poly-time solution to one NP-complete problem, they've automatically got one for all the NP-complete problems; that will also mean that P=NP.

A problem is NP-hard if and only if it's "at least as" hard as an NP-complete problem. The more conventional Traveling Salesman Problem of finding the shortest route is NP-hard, not strictly NP-complete.

I remember checking out Amit's Game Development page back when I wrote some games. He has a great sub-section on tiles that has most of what you want.