Approximation Algorithms Part II

Start Date: 10/25/2020

Course Type: Common Course

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Approximation algorithms, Part 2 This is the continuation of Approximation algorithms, Part 1. Here you will learn linear programming duality applied to the design of some approximation algorithms, and semidefinite programming applied to Maxcut. By taking the two parts of this course, you will be exposed to a range of problems at the foundations of theoretical computer science, and to powerful design and analysis techniques. Upon completion, you will be able to recognize, when faced with a new combinatorial optimization problem, whether it is close to one of a few known basic problems, and will be able to design linear programming relaxations and use randomized rounding to attempt to solve your own problem. The course content and in particular the homework is of a theoretical nature without any programming assignments. This is the second of a two-part course on Approximation Algorithms.

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Course Introduction

Approximation Algorithms Part II This course is the third part of a three-part series on approximations to matrices and applications of variance reduction to be a precondition for many applications in computer science and engineering. We will focus on the notion of perfect approximations, and discuss the problems of dimensionality of inequalities. We will also discuss the notion of approximate equality, and the problems of dimensionality of inequality. Approximation Algorithms Part I is the introductory course in optimization, in which we discuss and apply the concepts that are covered in the Powerplay and Incomplete functions, as well as in the Subprogression and Incomplete functions, as well as the inequality reduction and inequality reduction algorithms. We then proceed to a more advanced course in linear algebra and applications, in which we discuss more advanced topics in graph theory, and the power series and the gradient of functions. Course Overview: This course is designed to introduce you to computing with an introductory level approach, and to help you pick up the concepts that you need to know to apply to solve problems in computing. You will first install and use the free Cogent ( that is used by CERN, the European particle physics laboratory in Geneva, to program the computers that work at the CERN particle physics lab in Geneva. We will use an older version of Cogent (called Minitab) to program the computers that are in use at the

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Sparse approximation There are several algorithms that have been developed for solving sparse approximation problem.
Approximation algorithm In computer science and operations research, approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP-hard problems; since it is unlikely that there can ever be efficient polynomial-time exact algorithms solving NP-hard problems, one settles for polynomial-time sub-optimal solutions. Unlike heuristics, which usually only find reasonably good solutions reasonably fast, one wants provable solution quality and provable run-time bounds. Ideally, the approximation is optimal up to a small constant factor (for instance within 5% of the optimal solution). Approximation algorithms are increasingly being used for problems where exact polynomial-time algorithms are known but are too expensive due to the input size.
Approximation algorithm Not all approximation algorithms are suitable for all practical applications. They often use IP/LP/Semidefinite solvers, complex data structures or sophisticated algorithmic techniques which lead to difficult implementation problems. Also, some approximation algorithms have impractical running times even though they are polynomial time, for example O("n")
Approximation As another example, in order to accelerate the convergence rate of evolutionary algorithms, fitness approximation—that leads to build model of the fitness function to choose smart search steps—is a good solution.
CUR matrix approximation The CUR matrix approximation is not unique and there are multiple algorithms for computing one. One is ALGORITHMCUR.
Hardness of approximation Hardness of approximation complements the study of approximation algorithms by proving, for certain problems, a limit on the factors with which their solution can be efficiently approximated. Typically such limits show a factor of approximation beyond which a problem becomes NP-hard, implying that finding a polynomial time approximation for the problem is impossible unless NP=P. Some hardness of approximation results, however, are based on other hypotheses, a notable one among which is the unique games conjecture.
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.
Approximation algorithm NP-hard problems vary greatly in their approximability; some, such as the bin packing problem, can be approximated within any factor greater than 1 (such a family of approximation algorithms is often called a polynomial time approximation scheme or "PTAS"). Others are impossible to approximate within any constant, or even polynomial factor unless P = NP, such as the maximum clique problem.
Approximation algorithm For some approximation algorithms it is possible to prove certain properties about the approximation of the optimum result. For example, a "ρ"-approximation algorithm "A" is defined to be an algorithm for which it has been proven that the value/cost, "f"("x"), of the approximate solution "A"("x") to an instance "x" will not be more (or less, depending on the situation) than a factor "ρ" times the value, OPT, of an optimum solution.
Approximation algorithm NP-hard problems can often be expressed as integer programs (IP) and solved exactly in exponential time. Many approximation algorithms emerge from the linear programming relaxation of the integer program.
Stochastic approximation For this purpose, you can do experiments or run simulations to evaluate the performance of the system at given values of the parameters. Stochastic approximation algorithms have also been used in the social sciences to describe collective dynamics: fictitious play in learning theory and consensus algorithms can be studied using their theory
Approximation algorithm An ϵ-term may appear when an approximation algorithm introduces a multiplicative error and a constant error while the minimum optimum of instances of size "n" goes to infinity as "n" does. In this case, the approximation ratio is "c" ∓ "k" / OPT = "c" ∓ o(1) for some constants "c" and "k". Given arbitrary ϵ > 0, one can choose a large enough "N" such that the term "k" / OPT < ϵ for every "n ≥ N". For every fixed ϵ, instances of size "n < N" can be solved by brute force , thereby showing an approximation ratio — existence of approximation algorithms with a guarantee — of "c" ∓ ϵ for every ϵ > 0.
Approximation Approximation theory is a branch of mathematics, a quantitative part of functional analysis. Diophantine approximation deals with approximations of real numbers by rational numbers. Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain. However some known form may exist and may be able to represent the real form so that no significant deviation can be found. It also is used when a number is not rational, such as the number π, which often is shortened to 3.14159, or √2 to 1.414.
Approximation algorithm Inapproximability has been a fruitful area of research in computational complexity theory since the 1990 result of Feige, Goldwasser, Lovász, Safra and Szegedy on the inapproximability of Independent Set. After Arora et al. proved the PCP theorem a year later, it has now been shown that Johnson's 1974 approximation algorithms for Max SAT, Set Cover, Independent Set and Coloring all achieve the optimal approximation ratio, assuming P ≠ NP.
Approximation The type of approximation used depends on the available information, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.
Graph edit distance In addition to exact algorithms, a number of efficient approximation algorithms are
Hardness of approximation Since the early 1970s it was known that many optimization problems could not be solved in polynomial time unless P = NP, but in many of these problems the optimal solution could be efficiently approximated to a certain degree. In the 1970s, Teofilo F. Gonzalez and Sartaj Sahni began the study of hardness of approximation, by showing that certain optimization problems were NP-hard even to approximate to within a given approximation ratio. That is, for these problems, there is a threshold such that any polynomial-time approximation with approximation ratio beyond this threshold could be used to solve NP-complete problems in polynomial time. In the early 1990s, with the development of PCP theory, it became clear that many more approximation problems were hard to approximate, and that (unless P = NP) many known approximation algorithms achieved the best possible approximation ratio.
Approximation An approximation is anything that is similar but not exactly equal to something else.
Stochastic approximation Stochastic approximation methods are a family of iterative stochastic optimization algorithms that attempt to find zeroes or extrema of functions which cannot be computed directly, but only estimated via noisy observations. This situation is common, for instance, when taking noisy measurements of empirical data, or when computing parameters of a statistical model.
Approximation Approximation arises naturally in scientific experiments. The predictions of a scientific theory can differ from actual measurements. This can be because there are factors in the real situation that are not included in the theory. For example, simple calculations may not include the effect of air resistance. Under these circumstances, the theory is an approximation to reality. Differences may also arise because of limitations in the measuring technique. In this case, the measurement is an approximation to the actual value.