Computational complexity theoryIn 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 and is solvable by mechanical application of mathematical steps, such as an algorithm.
Complexity classIn computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". 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 resource like time or memory.
PPA (complexity)In computational complexity theory, PPA is a complexity class, standing for "Polynomial Parity Argument" (on a graph). Introduced by Christos Papadimitriou in 1994 (page 528), PPA is a subclass of TFNP. It is a class of search problems that can be shown to be total by an application of the handshaking lemma: any undirected graph that has a vertex whose degree is an odd number must have some other vertex whose degree is an odd number.
Computational problemIn theoretical computer science, a problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring"Given a positive integer n, find a nontrivial prime factor of n."is a computational problem that has a solution, as there are many known integer factorization algorithms. A computational problem can be viewed as a set of instances or cases together with a, possibly empty, set of solutions for every instance/case.
Reduction (complexity)In computability theory and computational complexity theory, a reduction is an algorithm for transforming one problem into another problem. A sufficiently efficient reduction from one problem to another may be used to show that the second problem is at least as difficult as the first. Intuitively, problem A is reducible to problem B, if an algorithm for solving problem B efficiently (if it exists) could also be used as a subroutine to solve problem A efficiently.NP (complexity)Unsolved problem in computer scienceP=?NP{\displaystyle {\mathsf {P\ {\overset {?}{=}}\ NP}}}More unsolved problems in computer scienceIn computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems.Co-NPIn computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement X is in the complexity class NP. The class can be defined as follows: a decision problem is in co-NP if and only if for every no-instance we have a polynomial-length "certificate" and there is a polynomial-time algorithm that can be used to verify any purported certificate.PLS (complexity)In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution to an optimization problem. The main characteristics of problems that lie in PLS are that the cost of a solution can be calculated in polynomial time and the neighborhood of a solution can be searched in polynomial time. Therefore it is possible to verify whether or not a solution is a local optimum in polynomial time.
NP-completenessIn computational complexity theory, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. Somewhat more precisely, a problem is NP-complete when: It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no".Each input to the problem is associated with a collection of short (polynomial size) solutions, which might or might not validly solve the input.
Computably enumerable setIn computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: There is an algorithm such that the set of input numbers for which the algorithm halts is exactly S.Or, equivalently, There is an algorithm that enumerates the members of S. That means that its output is a list of all the members of S: s1, s2, s3, ... .
Michael SipserMichael Fredric Sipser (born September 17, 1954) is an American theoretical computer scientist who has made early contributions to computational complexity theory. He is a professor of applied mathematics and was the dean of science at the Massachusetts Institute of Technology.
NP-hardnessIn computational complexity theory, a computational problem H is called NP-hard if, for every problem L which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from L to H. That is, assuming a solution for H takes 1 unit time, H's solution can be used to solve L in polynomial time.*As an Amazon Associate I earn from qualifying purchases.