Complexity theory in computation

Complexity Theory is a part of the theory of computation dealing with the the resources required during computation to solve a given problem. The most common resources are time (how many steps does it take to solve a problem) and space (how much memory does it take to solve a problem). Other resources can also be considered, such as how many parallel processors are needed to solve a problem in parallel. Complexity theory differs from computability theory, which deals with whether a problem can be solved at all, regardless of the resources required.

A single "problem" is an entire set of related questions, where each question is a finite-length string. For example, the problem FACTORIZE is: given an integer written in binary, return all of the prime factors of that number. A particular question is also called an instance of the problem.

The time complexity of a problem is the number of steps that it takes to solve an instance, as a function of the size of the instance. If an instance that is n bits long can be solved in n² steps, then we say it has a time complexity of n². Of course, the exact number of steps will depend on exactly what machine or language is being used. To avoid that problem, we generally use Big O notation. If a problem has time complexity O(n²) on one typical computer, then it will also have complexity O(n²) on most other computers, so this notation allows us to generalize away from the details of a particular computer.

Decision Problems

Much of complexity theory deals with decision problems. A decision problem is a problem where the answer is always YES/NO. For example, the problem IS-PRIME is: given an integer written in binary, return whether it is a prime number or not. A decision problem is equivalent to a language, which is a set of finite-length strings. For a given decision problem, the equivalent language is the set of all strings for which the answer is YES.

Decision problems are often considered because an arbitrary problem can always be reduced to a decision problem. For example, the problem HAS-FACTOR is: given integers n and k written in binary, return whether n has any prime factors less than k. If we can solve HAS-FACTOR with a certain amount of resources, then we can use that solution to solve FACTORIZE without much more resources. Just do a binary search on k until you find the smallest factor of n. Then divide out that factor, and repeat until you find all the factors.

Complexity theory often makes a distinction between YES answers and NO answers. For example, the set NP is defined as the set of problems where the YES instances can be checked quickly. The complement of a problem is one where all the YES and NO answers are swapped, such as IS-COMPOSITE for IS-PRIME. The set Co-NP is the set of problems where the NO instances can be checked quickly.

The P=NP Question

The set P is the set of decision problems that can be solved in polynomial time. The question of whether P is the same set as NP is the most important open question in all of theoretical computer science. There is even a $1,000,000 prize for solving it. (See Complexity classes P and NP and oracles).

Questions like this motivate the concepts of hard and complete. A set of problems X is hard for a set of problems Y if every problem in Y can be transformed easily into some problem in X with the same answer. The definition of "easily" is different in different contexts. The most important hard set is NP-Hard. Set X is complete for Y if it is hard for Y, and is also a subset of Y. The most important complete set is NP-Complete. See the articles on those two sets for more detail on the definition of "hard" and "complete".

Famous Complexity Classes

Here are some of the classes of decision problems considered in complexity theory, along with rough definitions:

P		Solvable in polynomial time (see Complexity classes P and NP)
NP		YES answers checkable in polynomial time (see Complexity classes P and NP)
Co-NP		NO answers checkable in polynomial time
NP-Complete		The hardest problems in NP
NP-Hard		Either NP-Complete or harder
#P		Count solutions to an NP problem
#P-Complete		The hardest problems in #P
NC?		Solvable efficiently on parallel computers
P-Complete?		The hardest problems in P to solve on parallel computers
PSPACE?		Solvable with polynomial memory and unlimited time
PSPACE-Complete?		The hardest problems in PSPACE
EXPTIME?		Solvable with exponential time
EXPSPACE?		Solvable with exponential memory and unlimited time
BQP		Solvable quickly on a quantum computer
ZPP?		Solvable by randomized algorithms (answer is always right, average running time is polynomial)
BPP		Solvable in polynomial time by randomized algorithms (answer is probably right)
RP?		Solvable in polynomial time by randomized algorithms (NO answer is probably right, YES is certainly right)