Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those complexity classes to each other.
A computational problem is understood to be a task that is in principle amenable to being solved by a computer, which is equivalent to stating that the problem may be solved by mechanical application of mathematical steps, such as an algorithm.
Measures of difficulty
A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used.
The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage.
Other complexity measures are also used, such as:
- The amount of communication (used in communication complexity)
- The number of logic gates in a circuit (used in circuit complexity)
- The number of processors (used in parallel computing)
One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do.
Related fields
Closely related fields in theoretical computer science are analysis of algorithms and computability theory.
A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem.
More precisely, it tries to classify problems that can or cannot be solved with appropriately restricted resources.
In turn, imposing restrictions on the available resources is what distinguishes computational complexity from computability theory: the latter theory asks what kind of problems can, in principle, be solved algorithmically.
See also
- 2-satisfiability - the problem of determining whether a collection of two-valued (Boolean or binary) variables with constraints on pairs of variables can be assigned values satisfying all the constraints.
- Advice (complexity)
- Algorithm
- Analysis of algorithms
- Category:Computational problems
- Complexity class
- Computability theory
- Computation
- Computer science
- Context of computational complexity
- Decision problem
- Descriptive complexity theory
- Formal language
- Game complexity
- List of complexity classes
- List of computability and complexity topics
- List of important publications in theoretical computer science
- List of unsolved problems in computer science
- Logic gate
- Mathematics
- NP-completeness
- Parallel computation with molecular-motor-propelled agents in nanofabricated networks
- Parallel computing
- Parameterized complexity
- Proof complexity
- Quantum complexity theory
- Structural complexity theory
- Transcomputational problem
- Theoretical computer science
- Theory of computation
External links
- Computational complexity theory @ Wikipedia
