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The P versus NP problem is a major unsolved problem in theoretical computer science. Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved. Here, quickly means an algorithm that solves the task and runs in polynomial time exists, meaning the task completion time varies as a polynomial ...
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of ...
Euler diagram for P, NP, NP-complete, and NP-hard set of problems (excluding the empty language and its complement, which belong to P but are not NP-complete) Main article: P versus NP problem The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time ), an algorithm ...
Under the assumption that P ≠ NP, the existence of problems within NP but outside both P and NP-complete was established by Ladner. [ 1] In computational complexity theory, NP ( nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where ...
Complexity class. In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity ". [1] The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded ...
The concept of NP-completeness was introduced in 1971 (see Cook–Levin theorem ), though the term NP-complete was introduced later. At the 1971 STOC conference, there was a fierce debate between the computer scientists about whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine.
Polynomial hierarchy. In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. [1] Each class in the hierarchy is contained within PSPACE.
In particular, we often show that a problem P is undecidable by showing that the halting problem reduces to P. The complexity classes P, NP and PSPACE are closed under (many-one, "Karp") polynomial-time reductions. The complexity classes L, NL, P, NP and PSPACE are closed under log-space reduction.