Explain Like I'm Five

Alright, part 2, let's get to NP.

Knowing that P means "in polynomial time", you might be tempted to think that NP means "in non-polynomial time" and while that kind of goes in the right direction, it means "in non-deterministic polynomial time". Explaining what non-deterministic calculations are would be a bit too complicated for an ELI5, so let's simplify a bit. A regular computer must make all decisions (for example which way to turn when calculating a shortest route between two points) based on the problem input alone. A non-deterministic computer can randomly guess. For judging complexity, we look at the case where it just happens to always guesses right. Even when guessing right, such a computer doesn't solve a problem immediately because it needs to make a number of guesses that depends on the input (for example the number of road junctions between our points). NP is the class of problems that a non-deterministic computer can solve in polynomial Time (O(n^a) for any a).

Obviously, we don't really have computers that always guess right, though quantum computers can get us a bit closer. But there are three important properties that let us understand NP problems in terms of regular computers:

1. a non-deterministic computer can do everything a regular computer can do (and more), so every problem that's part of P is also part of NP.
2. every problem that takes n guesses with x options for each guess can be simulated on a regular computer in O(x^n) steps by just trying all combinations of options and picking the best one. With some math, we can show that this is also true if we don't have n but O(n^a) guesses. Our base x might be different, but we can always find something with n in the exponent.
3. While finding a solution on a regular computer may need exponential time, we can always check if a solution is correct in polynomial time.

One important example for a problem in NP is finding the prime factors of a number which is why that is an important basic operation in cryptography. It's also an intuitive example for checking the result being easy. To check the result, we just need to multiply the factors together and see if we get our original number. Okay, technically we also need to check if each of the factors we get is really prime but as mentioned above, that's also doable in polynomial time.

Now for the important thing: we don't know if there is some shortcut that lets us simulate NP problems on a regular computer in polynomial time (even with a very high exponent) which would make NP equal to P.

What we do know is that there are some special problems (either from NP or even more complex) where every single problem from NP can be rephrased it as a combination of that special problem plus some extra work that's in P (for example converting our inputs and outputs to/from a format that fits L). Doing this rephrasing is absolutely mind-bending but there are clever computer scientists who have found a whole group of such problems. We call them NP-hard.

Why does this help us? Because finding a polynomial-time solution for just a single NP-hard problem would mean that by definition we can solve every single problem from NP by solving this polynomial-time NP-hard problem plus some polynomial-time extra work, so polynomial-time work overall. This would instantly make NP equal to P.

This leaves us with the definition of NP-complete. This is simply the class of problems that are both NP-hard and themselves in NP. This definition is useful for finding out if a problem is NP-hard but I think I've done enough damage to your 5-year-old brain.