Lecture 5 - Complexity of Games

The reason for NP’s fame is that it contains many of the problems that arise naturally in our lives. Technically, only decision problems, in which we must answer “yes” or “no” according to whether our input is a member of a particular set, are eligible for membership in NP. If it’s easy to verify that the answer to a particular decision problem is “yes”, then the problem is indeed in NP. On the other hand, if it’s easy to verify that the answer is “no”, then the problem is in a class called co-NP. Some problems can be in both classes, such as the primality problem. Since the invention of the AKS algorithm, we can determine in polynomial time whether a number is prime, making “yes” instances of the problem easy to verify. But the “no” instances are easy to verify as well, if we are provided with one of the factors of the number in question.