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NP-hard Problems
In 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 a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected, but unproven, that P≠NP, it is unlikely that any polynomial-time algorithms for NP-hard problems exist. A simple example of an NP-hard problem is the subset sum problem. Informally, if ''H'' is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are probably not NP-h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P Np Np-complete Np-hard
P, or p, is the sixteenth letter (alphabet), letter of the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet#Letter names, ''pee'' (pronounced ), plural ''pees''. History The Semitic languages, Semitic Pê (mouth), as well as the Greek language, Greek Π or π (Pi (letter), Pi), and the Etruscan language, Etruscan and Latin letters that developed from the former alphabet all symbolized , a voiceless bilabial plosive. Use in writing systems English In English orthography, represents the sound . A common Digraph (orthography), digraph in English is , which represents the sound , and can be used to transliterate ''phi'' in loanwords from Greek language, Greek. In German, the digraph is common, representing a labial affricate . Most English words beginning with are of foreign origin, primarily French, Latin and Greek; these languages pres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halting Problem
In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is ''Undecidable problem, undecidable'', meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. The problem comes up often in discussions of computability since it demonstrates that some functions are mathematically Definable set, definable but not Computable function, computable. A key part of the formal statement of the problem is a mathematical definition of a computer and program, usually via a Turing machine. The proof then shows, for any program that might determine whether programs halt, that a "pathological" program exists for which makes an incorrect determination. Specifically, is the program that, when called with some input, passes its own s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), adversarial behavior. More generally, cryptography is about constructing and analyzing Communication protocol, protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security (confidentiality, data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, Smart card#EMV, chip-based payment cards, digital currencies, password, computer passwords, and military communications. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Configuration Management
Configuration management (CM) is a management process for establishing and maintaining consistency of a product's performance, functional, and physical attributes with its requirements, design, and operational information throughout its life. The CM process is widely used by military engineering organizations to manage changes throughout the system lifecycle of complex systems, such as weapon systems, military vehicles, and information systems. Outside the military, the CM process is also used with IT service management as defined by ITIL, and with other domain models in the civil engineering and other industrial engineering segments such as roads, bridges, canals, dams, and buildings. Introduction CM applied over the life cycle of a system provides visibility and control of its performance, functional, and physical attributes. CM verifies that a system performs as intended, and is identified and documented in sufficient detail to support its projected life cycle. The CM proces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximate Computing
Approximate computing is an emerging paradigm for energy-efficient and/or high-performance design. It includes a plethora of computation techniques that return a possibly inaccurate result rather than a guaranteed accurate result, and that can be used for applications where an approximate result is sufficient for its purpose. One example of such situation is for a search engine where no exact answer may exist for a certain search query and hence, many answers may be acceptable. Similarly, occasional dropping of some frames in a video application can go undetected due to perceptual limitations of humans. Approximate computing is based on the observation that in many scenarios, although performing exact computation requires large amount of resources, allowing bounded approximation can provide disproportionate gains in performance and energy, while still achieving acceptable result accuracy. For example, in ''k''-means clustering algorithm, allowing only 5% loss in classification ac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-intermediate
In computational complexity, problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such problems is called NPI. Ladner's theorem, shown in 1975 by Richard E. Ladner, is a result asserting that, if P ≠ NP, then NPI is not empty; that is, NP contains problems that are neither in P nor NP-complete. Since it is also true that if NPI problems exist, then P ≠ NP, it follows that P = NP if and only if NPI is empty. Under the assumption that P ≠ NP, Ladner explicitly constructs a problem in NPI, although this problem is artificial and otherwise uninteresting. It is an open question whether any "natural" problem has the same property: Schaefer's dichotomy theorem provides conditions under which classes of constrained Boolean satisfiability problems cannot be in NPI. Some problems that are considered good candidates for being NP-intermediate are the graph isomorphism problem, and decision versions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-equivalent
In computational complexity theory, the complexity class NP-equivalent is the set of function problems that are both NP-easy and NP-hard. NP-equivalent is the analogue of NP-complete for function problems. For example, the problem FIND-SUBSET-SUM is in NP-equivalent. Given a set of integers, FIND-SUBSET-SUM is the problem of finding some nonempty subset of the integers that adds up to zero (or returning the empty set if there is no such subset). This optimization problem is similar to the decision problem SUBSET-SUM. Given a set of integers, SUBSET-SUM is the problem of finding whether there exists a subset summing to zero. SUBSET-SUM is NP-complete. To show that FIND-SUBSET-SUM is NP-equivalent, we must show that it is both NP-hard and NP-easy. Clearly it is NP-hard. If we had a black box that solved FIND-SUBSET-SUM in unit time, then it would be easy to solve SUBSET-SUM. Simply ask the black box to find the subset that sums to zero, then check whether it returned a nonem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-easy
In complexity theory, the complexity class NP-easy is the set of function problems that are solvable in polynomial time by a deterministic Turing machine with an oracle for some decision problem in NP. In other words, a problem X is NP-easy if and only if there exists some problem Y in NP such that X is polynomial-time Turing reducible to Y., p. 117, 120. This means that given an oracle for Y, there exists an algorithm that solves X in polynomial time (possibly by repeatedly using that oracle). NP-easy is another name for FPNP (see the function problem article) or for FΔ2P (see the polynomial hierarchy article). An example of an NP-easy problem is the problem of sorting a list of strings. The decision problem "is string A greater than string B" is in NP. There are algorithms such as quicksort that can sort the list using only a polynomial number of calls to the comparison routine, plus a polynomial amount of additional work. Therefore, sorting is NP-easy. There are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
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 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 logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSPACE-complete
In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length (PSPACE, polynomial space) and if every other problem that can be solved in polynomial space can be Polynomial-time reduction, transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, the class of decision problems solvable in polynomial space, because a solution to any one such problem could easily be used to solve any other problem in PSPACE. Problems known to be PSPACE-complete include determining properties of regular expressions and context-sensitive grammars, determining the truth of quantified Boolean formula problem, quantified Boolean formulas, step-by-step changes between solutions of combinatorial optimization problems, and many puzzles and games. Theory A problem is defined to be PSPACE-complete if it can be solved using a polynomial amount o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSPACE
In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. Formal definition If we denote by SPACE(''f''(''n'')), the set of all problems that can be solved by Turing machines using ''O''(''f''(''n'')) space for some function ''f'' of the input size ''n'', then we can define PSPACE formally asArora & Barak (2009) p.81 :\mathsf = \bigcup_ \mathsf(n^k). It turns out that allowing the Turing machine to be nondeterministic does not add any extra power. Because of Savitch's theorem,Arora & Barak (2009) p.85 NPSPACE is equivalent to PSPACE, essentially because a deterministic Turing machine can simulate a nondeterministic Turing machine without needing much more space (even though it may use much more time).Arora & Barak (2009) p.86 Also, the complements of all problems in PSPACE are also in PSPACE, meaning that co-PSPACE PSPACE. Relation among other classes The following re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
True Quantified Boolean Formula
In computational complexity theory, the language TQBF is a formal language consisting of the true quantified Boolean formulas. A (fully) quantified Boolean formula is a formula in Quantification (logic), quantified propositional logic (also known as Second-order propositional logic) where every variable is quantified (or Bound variable, bound), using either existential quantification, existential or universal quantification, universal quantifiers, at the beginning of the sentence. Such a formula is equivalent to either true or false (since there are no Bound variable, free variables). If such a formula evaluates to true, then that formula is in the language TQBF. It is also known as QSAT (Quantified Boolean satisfiability problem, SAT). Overview In computational complexity theory, the quantified Boolean formula problem (QBF) is a generalization of the Boolean satisfiability problem in which both existential quantification, existential quantifiers and universal quantification, univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
