1. Section 14.3 Complexity Classes
A complexity class is a set of problems related by how much time or space algorithms take to
solve them. Complexity classes are usually defined for decision problems, which are
problems that ask a question with a yes/no answer. Here are some fundamental complexity
classes.
The class P
The set of problems that can be solved by deterministic algorithms with worst-case running
times of polynomial order.
The class NP
The set of problems for which solutions can be verified by deterministic algorithms with
worst-case running times of polynomial order. (The N stands for nondeterministic guess and
the P stands for polynomial time to verify that the guess is a solution.)
The class PSPACE
The set of problems that can be solved by deterministic algorithms where the worst-case
number of memory cells needed is of polynomial order.
Intractable Problems
The set of problems that are not in P.
NP-Completeness
Polynomially Reducible, NP-Competeness, CNF-Satisfiability, Cook’s Theorem, Algorithm
to find NP-complete problem, Restricting CNF-Satisfiability.

2. Example
The subset sum problem: given a positive integer A and a set B = {p1, ..., pn} of
positive integers, is there a subset of B whose sum is A?
Discussion: A simple deterministic algorithm to solve the problem would go through
all the subsets of B looking for a subset whose elements summed to A. Since there
are 2n subsets of B, this algorithm does not take polynomial time. But any subset can
be checked with at most n – 1 additions.Thus there is a O(n) algorithm to do the
checking. So the problem is in NP. The problem has been shown to be NP-complete.