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MAXEkSAT
MAXEkSAT is a problem in computational complexity theory that is a maximization version of the Boolean satisfiability problem 3SAT. In MAXEkSAT, each clause has exactly ''k'' literals, each with distinct variables, and is in conjunctive normal form. These are called k-CNF formulas. The problem is to determine the maximum number of clauses that can be satisfied by a truth assignment to the variables in the clauses. We say that an algorithm ''A'' provides an ''α''- approximation to MAXEkSAT if, for some fixed positive ''α'' less than or equal to 1, and every kCNF formula ''φ'', ''A'' can find a truth assignment to the variables of ''φ'' that will satisfy at least an ''α''-fraction of the maximum number of satisfiable clauses of ''φ''. Because the NP-hard ''k''-SAT problem (for ''k'' ≥ 3) is equivalent to determining if the corresponding MAXEkSAT instance has a value equal to the number of clauses, MAXEkSAT must also be NP-hard, meaning that there is no polynomial time algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Satisfiability Problem
In computational complexity theory, the maximum satisfiability problem (MAX-SAT) is the problem of determining the maximum number of clauses, of a given Boolean formula in conjunctive normal form, that can be made true by an assignment of truth values to the variables of the formula. It is a generalization of the Boolean satisfiability problem, which asks whether there exists a truth assignment that makes all clauses true. Example The conjunctive normal form formula : (x_0\lor x_1)\land(x_0\lor\lnot x_1)\land(\lnot x_0\lor x_1)\land(\lnot x_0\lor\lnot x_1) is not satisfiable: no matter which truth values are assigned to its two variables, at least one of its four clauses will be false. However, it is possible to assign truth values in such a way as to make three out of four clauses true; indeed, every truth assignment will do this. Therefore, if this formula is given as an instance of the MAX-SAT problem, the solution to the problem is the number three. Hardness The MAX-SAT problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairwise Independence
In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not mutually independent. Pairwise independent random variables with finite variance are uncorrelated. A pair of random variables ''X'' and ''Y'' are independent if and only if the random vector (''X'', ''Y'') with joint cumulative distribution function (CDF) F_(x,y) satisfies :F_(x,y) = F_X(x) F_Y(y), or equivalently, their joint density f_(x,y) satisfies :f_(x,y) = f_X(x) f_Y(y). That is, the joint distribution is equal to the product of the marginal distributions. Unless it is not clear in context, in practice the modifier "mutual" is usually dropped so that independence means mutual independence. A statement such as " ''X'', ''Y'', ''Z'' are independent random variables" means that ''X'', ''Y'', ''Z'' are mutua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Of BCH Is An Independent Source
A certain family of BCH codes have a particularly useful property, which is that treated as linear operators, their dual operators turns their input into an \ell-wise independent source. That is, the set of vectors from the input vector space are mapped to an \ell-wise independent source. The proof of this fact below as the following Lemma and Corollary is useful in derandomizing the algorithm for a 1-2^-approximation to MAXEkSAT MAXEkSAT is a problem in computational complexity theory that is a maximization version of the Boolean satisfiability problem 3SAT. In MAXEkSAT, each clause has exactly ''k'' literals, each with distinct variables, and is in conjunctive normal form .... Lemma Let C\subseteq F_2^n be a linear code such that C^\perp has distance greater than \ell +1. Then C is an \ell-wise independent source. Proof of lemma It is sufficient to show that given any k \times l matrix ''M'', where ''k'' is greater than or equal to ''l'', such that the rank of ''M'' is ' ... [...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 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 computationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Satisfiability Problem
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called ''satisfiable''. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is ''unsatisfiable''. For example, the formula "''a'' AND NOT ''b''" is satisfiable because one can find the values ''a'' = TRUE and ''b'' = FALSE, which make (''a'' AND NOT ''b'') = TRUE. In contrast, "''a'' AND NOT ''a''" is unsatisfiable. SAT is the first problem that was proved to be NP-complete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Complexity
In logic and theoretical computer science, and specifically proof theory and computational complexity theory, proof complexity is the field aiming to understand and analyse the computational resources that are required to prove or refute statements. Research in proof complexity is predominantly concerned with proving proof-length lower and upper bounds in various propositional proof systems. For example, among the major challenges of proof complexity is showing that the Frege system, the usual propositional calculus, does not admit polynomial-size proofs of all tautologies. Here the size of the proof is simply the number of symbols in it, and a proof is said to be of polynomial size if it is polynomial in the size of the tautology it proves. Systematic study of proof complexity began with the work of Stephen Cook and Robert Reckhow (1979) who provided the basic definition of a propositional proof system from the perspective of computational complexity. Specifically Cook and Reckhow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameterized Complexity
In computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to ''multiple'' parameters of the input or output. The complexity of a problem is then measured as a function of those parameters. This allows the classification of NP-hard problems on a finer scale than in the classical setting, where the complexity of a problem is only measured as a function of the number of bits in the input. The first systematic work on parameterized complexity was done by . Under the assumption that P ≠ NP, there exist many natural problems that require superpolynomial running time when complexity is measured in terms of the input size only, but that are computable in a time that is polynomial in the input size and exponential or worse in a parameter . Hence, if is fixed at a small value and the growth of the function over is relatively small then such p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Unsolved Problems In Computer Science
This article is a list of notable unsolved problems in computer science. A problem in computer science is considered unsolved when no solution is known, or when experts in the field disagree about proposed solutions. Computational complexity * P versus NP problem * What is the relationship between BQP and NP? * NC = P problem * NP = co-NP problem * P = BPP problem * P = PSPACE problem * L = NL problem * PH = PSPACE problem * L = P problem * L = RL problem * Unique games conjecture * Is the exponential time hypothesis true? ** Is the strong exponential time hypothesis (SETH) true? * Do one-way functions exist? ** Is public-key cryptography possible? * Log-rank conjecture Polynomial versus nondeterministic-polynomial time for specific algorithmic problems * Can integer factorization be done in polynomial time on a classical (non-quantum) computer? * Can the discrete logarithm be computed in polynomial time on a classical (non-quantum) computer? * Can the shortest vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Computability And Complexity Topics
This is a list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with what can be computed, in principle. Computational complexity theory deals with how hard computations are, in quantitative terms, both with upper bounds (algorithms whose complexity in the worst cases, as use of computing resources, can be estimated), and from below (proofs that no procedure to carry out some task can be very fast). For more abstract foundational matters, see the list of mathematical logic topics. See also list of algorithms, list of algorithm general topics. Calculation *Lookup table ** Mathematical table **Multiplication table **Generating trigonometric tables * History of computers *Multiplication algorithm ** Peasant multiplication *Division by two *Exponentiating by squaring * Addition chain **Scholz conjecture *Presburger arithmetic Computability theory: models of computation * Arithmetic circuits *Algorit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Complexity Classes
This is a list of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics. Many of these classes have a 'co' partner which consists of the complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...s of all languages in the original class. For example, if a language L is in NP then the complement of L is in co-NP. (This does not mean that the complement of NP is co-NP—there are languages which are known to be in both, and other languages which are known to be in neither.) "The hardest problems" of a class refer to problems which belong to the class such that every other problem of that class can be reduced to it. Furthermore, the reduction is also a problem of the given class, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Complexity Theory
Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic. This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow "natural" and not tied to the specific abstract machines used to define them. Specifically, each logical system produces a set of queries expressible in it. The queries – when restricted to finite structures – correspond to the computational problems of traditional complexity theory. The first main result of descriptive complexity was Fagin's theorem, shown by Ronald Fagin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Context Of Computational Complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem. The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory. Both areas are highly related, as the complexity of an algorithm is always an upper bound on the complexity of the problem solved by this algorithm. Moreover, for designing efficient algorithms, it is often fundamental to compare the complexity of a specific algorithm to the complexity of the problem to be solved. Also, in most cases, the only thing that is known about the complexity of a problem is that it is lower than the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
