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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 ...
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 ...
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.
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 ...
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 ...
The relationship between BPP and NP is unknown: it is not known whether BPP is a subset of NP, NP is a subset of BPP or neither. If NP is contained in BPP, which is considered unlikely since it would imply practical solutions for NP-complete problems, then NP = RP and PH ⊆ BPP. [4] It is known that RP is a subset of BPP, and BPP is a subset ...
If P and NP are different, then there exist decision problems in the region of NP that fall between P and the NP-complete problems. (If P and NP are the same class, then NP-intermediate problems do not exist because in this case every NP-complete problem would fall in P, and by definition, every problem in NP can be reduced to an NP-complete ...
The Boolean satisfiability problem (SAT) is, given a formula, to check whether it is satisfiable. This decision problem is of central importance in many areas of computer science, including theoretical computer science, complexity theory, [3] [4] algorithmics, cryptography [5] [6] and artificial intelligence. [7] [additional citation (s) needed]