|
Polynomial Time Approximation Scheme
In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter and produces a solution that is within a factor of being optimal (or for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most , with being the length of the shortest tour.Sanjeev Arora, Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems, Journal of the ACM 45(5) 753–782, 1998. The running time of a PTAS is required to be polynomial in the problem size for every fixed ε, but can be different for different ε. Thus an algorithm running in time or even counts as a PTAS. Variants Deterministic A practical problem with PTAS algorithms is that the exponent of the polynomial coul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Gerhard J
Gerhard is a name of Germanic origin and may refer to: Given name * Gerhard (bishop of Passau) (fl. 932–946), German prelate * Gerhard III, Count of Holstein-Rendsburg (1292–1340), German prince, regent of Denmark * Gerhard Barkhorn (1919–1983), German World War II flying ace * Gerhard Berger (born 1959), Austrian racing driver * Gerhard Boldt (1918–1981), German soldier and writer * Gerhard de Beer (born 1994), South African football player * Gerhard Diephuis (1817–1892), Dutch jurist * Gerhard Domagk (1895–1964), German pathologist and bacteriologist and Nobel Laureate * Gerhard Dorn (c.1530–1584), Flemish philosopher, translator, alchemist, physician and bibliophile * Gerhard Ertl (born 1936), German physicist and Nobel Laureate * Gerhard Fieseler (1896–1987), German World War I flying ace * Gerhard Flesch (1909–1948), German Nazi Gestapo and SS officer executed for war crimes * Gerhard Gentzen (1909–1945), German mathematician and logician * Gerhard A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Marek Karpinski
Marek Karpinski at the Hausdorff Center for Mathematics, Excellence Cluster is a and known for his research in the theory of s and their applications, , |
|
Approximation-preserving Reduction
In computability theory and computational complexity theory, especially the study of approximation algorithms, an approximation-preserving reduction is an algorithm for transforming one optimization problem into another problem, such that the distance of solutions from optimal is preserved to some degree. Approximation-preserving reductions are a subset of more general reductions in complexity theory; the difference is that approximation-preserving reductions usually make statements on approximation problems or optimization problems, as opposed to decision problems. Intuitively, problem A is reducible to problem B via an approximation-preserving reduction if, given an instance of problem A and a (possibly approximate) solver for problem B, one can convert the instance of problem A into an instance of problem B, apply the solver for problem B, and recover a solution for problem A that also has some guarantee of approximation. Definition Unlike reductions on decision problems, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
L-reduction
In computer science, particularly the study of approximation algorithms, an L-reduction ("''linear reduction''") is a transformation of optimization problems which linearly preserves approximability features; it is one type of approximation-preserving reduction. L-reductions in studies of approximability of optimization problems play a similar role to that of Polynomial-time reduction, polynomial reductions in the studies of Computational complexity theory, computational complexity of decision problems. The term ''L reduction'' is sometimes used to refer to log-space reductions, by analogy with the complexity class L (complexity), L, but this is a different concept. Definition Let A and B be optimization problem In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goo ...s and cA and cB their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
PTAS Reduction
In computational complexity theory, a PTAS reduction is an approximation-preserving reduction that is often used to perform reductions between solutions to optimization problems. It preserves the property that a problem has a polynomial time approximation scheme (PTAS) and is used to define completeness for certain classes of optimization problems such as APX. Notationally, if there is a PTAS reduction from a problem A to a problem B, we write \text \leq_ \text. With ordinary polynomial-time many-one reductions, if we can describe a reduction from a problem A to a problem B, then any polynomial-time solution for B can be composed with that reduction to obtain a polynomial-time solution for the problem A. Similarly, our goal in defining PTAS reductions is so that given a PTAS reduction from an optimization problem A to a problem B, a PTAS for B can be composed with the reduction to obtain a PTAS for the problem A. Definition Formally, we define a PTAS reduction from A to B using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Randomized Algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables. There is a distinction between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for example Quicksort), and algorithms which have a chance of producing an incorrect result ( Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem. In common practice, randomized alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Parameterized Approximation Algorithm
A parameterized approximation algorithm is a type of algorithm that aims to find approximate solutions to NP-hard optimization problems in polynomial time in the input size and a function of a specific parameter. These algorithms are designed to combine the best aspects of both traditional approximation algorithms and fixed-parameter tractability. In traditional approximation algorithms, the goal is to find solutions that are at most a certain factor away from the optimal solution, known as an -approximation, in polynomial time. On the other hand, parameterized algorithms are designed to find exact solutions to problems, but with the constraint that the running time of the algorithm is polynomial in the input size and a function of a specific parameter . The parameter describes some property of the input and is small in typical applications. The problem is said to be fixed-parameter tractable (FPT) if there is an algorithm that can find the optimum solution in f(k)n^ time, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Polylogarithmic Function
In mathematics, a polylogarithmic function in is a polynomial in the logarithm of , : a_k (\log n)^k + a_ (\log n)^ + \cdots + a_1(\log n) + a_0. The notation is often used as a shorthand for , analogous to for . In computer science, polylogarithmic functions occur as the order of time for some data structure In computer science, a data structure is a data organization and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the relationships amo ... operations. Additionally, the exponential function of a polylogarithmic function produces a function with quasi-polynomial growth, and algorithms with this as their time complexity are said to take quasi-polynomial time. All polylogarithmic functions of are for every exponent (for the meaning of this symbol, see small o notation), that is, a polylogarithmic function grows more slowly than any positive exponent. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Time Complexity
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
APX-hard
In computational complexity theory, the class APX (an abbreviation of "approximable") is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant (or constant-factor approximation algorithms for short). In simple terms, problems in this class have efficient algorithms that can find an answer within some fixed multiplicative factor of the optimal answer. An approximation algorithm is called an f(n)-approximation algorithm for input size n if it can be proven that the solution that the algorithm finds is at most a multiplicative factor of f(n) times worse than the optimal solution. Here, f(n) is called the ''approximation ratio''. Problems in APX are those with algorithms for which the approximation ratio f(n) is a constant c. The approximation ratio is conventionally stated greater than 1. In the case of minimization problems, f(n) is the found solution's score divided by the optimum solution's score, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |