An Overview of P versus NP

In Chapter 8, Dynamic Programming I, we demonstrated the significant gains in efficiency that dynamic programming can offer over other approaches, but it may not yet be clear how dramatic the difference can be. It is important to appreciate the extent to which the complexity of certain problems will scale as the input bounds increase because then we can understand the situations in which DP is not just preferable, but necessary.

Consider the following problem:

"Given the terms and operators of a Boolean formula, determine whether or not it evaluates to TRUE."

Take a look at the following example:

(0 OR 1)  —> TRUE

(1 AND 0) —> FALSE

(1 NOT 1) —> FALSE

(1 NOT 0) AND (0 NOT 1) —> TRUE

This problem is conceptually very simple to solve. All that is required to get the correct result is a linear evaluation of the given formula. However, imagine that, instead, the problem was stated this way:

"Given the variables...