Computational complexity theory

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.

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 their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and 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. The P versus NP problem, one of the seven Millennium Prize Problems,^[1] is part of the field of computational complexity.

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, computational complexity theory 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 kinds of problems can, in principle, be solved algorithmically.

The type of computational problem: The most commonly used problems are decision problems. However, complexity classes can be defined based on , counting problems, optimization problems, promise problems, etc.

function problems

The model of computation: The most common model of computation is the deterministic Turing machine, but many complexity classes are based on non-deterministic Turing machines, , quantum Turing machines, monotone circuits, etc.

Boolean circuits

The resource (or resources) that is being bounded and the bound: These two properties are usually stated together, such as "polynomial time", "logarithmic space", "constant depth", etc.

Continuous complexity theory[edit]

Continuous complexity theory can refer to complexity theory of problems that involve continuous functions that are approximated by discretizations, as studied in numerical analysis. One approach to complexity theory of numerical analysis^[17] is information based complexity.

Continuous complexity theory can also refer to complexity theory of the use of analog computation, which uses continuous dynamical systems and differential equations.^[18] Control theory can be considered a form of computation and differential equations are used in the modelling of continuous-time and hybrid discrete-continuous-time systems.^[19]

Wuppuluri, Shyam; Doria, Francisco A., eds. (2020), , World Scientific, doi:10.1142/11270, ISBN 978-981-12-0006-9, S2CID 198790362

Unravelling Complexity: The Life and Work of Gregory Chaitin

; Barak, Boaz (2009), Computational Complexity: A Modern Approach, Cambridge University Press, ISBN 978-0-521-42426-4, Zbl 1193.68112

Arora, Sanjeev

Downey, Rod; (1999), Parameterized complexity, Monographs in Computer Science, Berlin, New York: Springer-Verlag, ISBN 9780387948836

Fellows, Michael

Du, Ding-Zhu; Ko, Ker-I (2000), Theory of Computational Complexity, John Wiley & Sons, 978-0-471-34506-0

ISBN

; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences (1st ed.). New York: W. H. Freeman and Company. ISBN 9780716710455. MR 0519066. OCLC 247570676.

Garey, Michael R.

(2008), Computational Complexity: A Conceptual Perspective, Cambridge University Press

Goldreich, Oded

, ed. (1990), Handbook of theoretical computer science (vol. A): algorithms and complexity, MIT Press, ISBN 978-0-444-88071-0

van Leeuwen, Jan

(1994), Computational Complexity (1st ed.), Addison Wesley, ISBN 978-0-201-53082-7

Papadimitriou, Christos

(2006), Introduction to the Theory of Computation (2nd ed.), USA: Thomson Course Technology, ISBN 978-0-534-95097-2