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Algorithm for transforming one optimization problem into another
complexity theory, especially the study of approximation algorithms, an approximation-preserving reduction is an algorithm for transforming one optimization
Approximation-preserving reduction
Approximation-preserving_reduction
preserves approximability features; it is one type of approximation-preserving reduction. L-reductions in studies of approximability of optimization problems
L-reduction
Transformation of one computational problem to another
solutions to instances of problem A. Approximation-preserving reductions are often used to prove hardness of approximation results: if some optimization problem
Reduction_(complexity)
Approximation-preserving reduction
reduction is an approximation-preserving reduction that is often used to perform reductions between solutions to optimization problems. It preserves the
PTAS_reduction
Topics referred to by the same term
approximation scheme, an approximation algorithm in computer science Pesetas, Spanish currency PTAS reduction, an approximation-preserving reduction in
PTAS
exponential with a low base. Approximation-preserving reduction APX is the class of problems with some constant-factor approximation algorithm Heuristic algorithm
Exact_algorithm
Class of algorithms that find approximate solutions to optimization problems
constant-factor approximation algorithm Approximation-preserving reduction Exact algorithm Bernard., Shmoys, David (2011). The design of approximation algorithms
Approximation_algorithm
solvable in polynomial time. Approximation-preserving reduction Optimization problem Approximation algorithm PTAS reduction Demaine, Erik (Fall 2014). "Algorithmic
Gap_reduction
Class of problems in computer science
have a PTAS unless P=NP. This can be proved by showing an approximation-preserving reduction from MAX 3-SAT-3 to GISMP2. The following greedy algorithm
Interval_scheduling
NP-hard problem in combinatorial optimization
the Wayback Machine Orponen, P.; Mannila, H. (1987). On approximation preserving reductions: Complete problems and robust measures' (Report). Department
Travelling_salesman_problem
Complexity class of approximable problems
their value, hence the exponential factor. Approximation-preserving reduction Complexity class Approximation algorithm Max/min CSP/Ones classification
APX
Projection of data onto lower-dimensional manifolds
similar distribution. Uniform manifold approximation and projection (UMAP) is a nonlinear dimensionality reduction technique. It works in a similar fashion
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Process of reducing the number of random variables under consideration
necessarily preserve densities or distances well. Uniform manifold approximation and projection (UMAP) is a nonlinear dimensionality reduction technique
Dimensionality_reduction
Computational geometry and optimization concept
geometry and approximation algorithms, a coreset is a small, possibly weighted subset of an input point set that approximately preserves the value of
Coreset
Type of approximation algorithm
APX-hardness is commonly shown via PTAS reduction or AP-reduction. Parameterized approximation scheme, an approximation scheme that runs in FPT time Sanjeev
Polynomial-time approximation scheme
Polynomial-time_approximation_scheme
Optimization problem
to within a factor O(log n). This factor is tight, via an approximation-preserving reduction from the set cover problem. If we assume distances between
Optimal_facility_location
Algorithmic technique using hashing
(LSH); or data-dependent methods, such as locality-preserving hashing (LPH). Locality-preserving hashing was initially devised as a way to facilitate
Locality-sensitive_hashing
Minimum-cost tree with exactly k vertices
NP-hardness reduction for the k-minimum spanning tree problem preserves the weight of all solutions, it also preserves the hardness of approximation of the
K-minimum_spanning_tree
Type of algorithm
A parameterized approximation algorithm is a type of algorithm that aims to find approximate solutions to NP-hard optimization problems in polynomial
Parameterized approximation algorithm
Parameterized_approximation_algorithm
Vertices whose removal breaks all cycles
The existence of an approximation preserving L-reduction from the vertex cover problem to it; Existing constant-factor approximation algorithms. The best
Feedback_vertex_set
Subunit of a computational problem
derived from the reduction will depend on how well the gap is preserved. Trevisan et al. (2000) formalize the problem of finding gap-preserving gadgets, for
Gadget_(computer_science)
Technique to reduce dimensionality of points in Euclidean space
Dimensionality reduction is often used to reduce the problem of managing and manipulating large data sets. Dimensionality reduction techniques generally
Random_projection
Complexity class
polynomial time. In some cases parsimonious reductions, a more specific type of reduction that preserves the exact number of solutions, are used. #P-complete
♯P-complete
Distance-preserving mathematical transformation
an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective
Isometry
Lists of values of mathematical functions
combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, Padé approximation, and typically for higher or
Trigonometric_table
Simplifying model for electrical grids
By applying network reduction techniques transmission system operators can significantly reduce simulation time while preserving the essential dynamic
Power_system_reduction
Algorithms for calculating square roots
these algorithms typically construct a series of increasingly accurate approximations. Most square root computation methods are iterative: after choosing
Square_root_algorithms
polynomial-time approximation scheme (FPRAS), even assuming that each variable occurs in at most 6 clauses, but that a fully polynomial-time approximation scheme
♯SAT
Subfield of mathematical optimization
some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in
Combinatorial_optimization
Mathematical and computational problem
problem can be produced with sophisticated algorithms. In addition, many approximation algorithms exist. For example, the first fit algorithm provides a fast
Bin_packing_problem
Degradation of AI models trained on synthetic data
discourage over-memorisation and maintain output variety while preserving task performance. Preserving output diversity in this way has also been reported to
Model_collapse
Subset of a graph's vertices, including at least one endpoint of every edge
approximation-preserving way: The Independent Set problem has no constant-factor approximation unless P = NP. Approximation algorithm: APPROXIMATION-VERTEX-COVER(G)
Vertex_cover
Gender-affirming surgery undertaken by transgender people to alter their voice
Mehmet Mazhar (2010). "Laser Reduction Glottoplasty for Managing Androphonia After Failed Cricothyroid Approximation Surgery". Journal of Voice. 24
Transgender_voice_surgery
Subset of a graph's nodes such that all other nodes link to at least one
and vice versa. Moreover, the reductions preserve the approximation ratio: for any α, a polynomial-time α-approximation algorithm for minimum dominating
Dominating_set
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
non-overlapping way. Because f is locally area-preserving, this non-overlapping property would make it area-preserving for all of S, so the area of f (S) would
Minkowski's_theorem
Surgery to increase voice pitch
thyroplasty that is distinct in critically avoiding a cricothyroid approximation. Instead, as a form of open laryngoplasty a partial laryngectomy of
Feminization_laryngoplasty
Unsolved problem in computational complexity theory
complexity. It has broad applications in the theory of hardness of approximation. If the unique games conjecture is true and P ≠ NP, then for many important
Unique_games_conjecture
Approximation of a mathematical set
a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the lower and the upper approximation of the original
Rough_set
Noise removal process during image processing
variation denoising is a remarkably effective edge-preserving filter, i.e., simultaneously preserving edges whilst smoothing away noise in flat regions
Total_variation_denoising
Matrix of partial derivatives of a vector-valued function
product Jf(x) ⋅ h is another displacement vector, that is the best linear approximation of the change of f along h in a neighborhood of x, if f(x) is differentiable
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Type of data compression
compression, is the class of data compression methods that uses inexact approximations and partial data discarding to represent the content. These techniques
Lossy_compression
Representation learning technique
extraction Dimensionality reduction Word embedding Neural network Reinforcement learning Metric space#Metric embeddings and approximations Bengio, Yoshua; Ducharme
Embedding_(machine_learning)
Variant of the traveling salesman problem
However, although this transformation preserves the optimal solution, it does not preserve the quality of approximations to that solution. If the graph is
Bottleneck traveling salesman problem
Bottleneck_traveling_salesman_problem
Machine learning technique useful for dimensionality reduction
two-dimensional) representation of a higher-dimensional data set while preserving the topological structure of the data. For example, a data set with p
Self-organizing_map
Changing the resolution of a digital image
an approximation to the sinc method, yields better results. Bicubic interpolation can be regarded as a computationally efficient approximation to Lanczos
Image_scaling
Method of data analysis
component analysis Low-rank approximation Matrix decomposition Non-negative matrix factorization Nonlinear dimensionality reduction Oja's rule Point distribution
Principal_component_analysis
Problem in computer science
is useful because MAX-3SAT(B) can often be used to obtain a PTAS-preserving reduction in a way that MAX-3SAT cannot. Proofs for explicit values of B include:
MAX-3SAT
Dimensionality reduction algorithm
computationally expensive to build than standard DMD models. Measure-preserving EDMD: Measure-preserving extended DMD (mpEDMD) offers a Galerkin method whose eigendecomposition
Dynamic_mode_decomposition
Image denoising algorithm
a factor of 50 while preserving comparable quality of the result. Anisotropic diffusion Digital image processing Noise reduction Nonlocal operator Signal
Non-local_means
Consonant sound change
changes: voicing, approximation, and vocalization.[clarification needed] [t] > [d] (voicing, example in Korean) [d] > [ð] (approximation, example in Spanish)
Lenition
Curve used in computer graphics and related fields
} Degree reduction can only be done exactly when the curve in question is originally elevated from a lower degree. A number of approximation algorithms
Bézier_curve
On short connecting nets with added points
instance of the metric Steiner tree problem; the transformation preserves the approximation factor. While the Euclidean version admits a PTAS, it is known
Steiner_tree_problem
Feature enhancement algorithm in imaging science
separable filter. The difference of Gaussians can be thought of as an approximation of the Mexican hat kernel function used for the Laplacian of the Gaussian
Difference_of_Gaussians
Type of activation function
paper also introduces a few faster approximations for GELU. The first approximation follows from an approximation for Φ ( x ) {\displaystyle \Phi (x)}
Rectified_linear_unit
complex numbers Gamma function: Lanczos approximation Spouge's approximation — modification of Stirling's approximation; easier to apply than Lanczos AGM method
List of numerical analysis topics
List_of_numerical_analysis_topics
Edges that hit all cycles in a graph
the Karp–Lawler reduction from vertex cover to feedback arc set, which preserves the quality of approximations. By a different reduction, the maximum acyclic
Feedback_arc_set
Theorem in mathematics
an inverse function. The essential idea is that if the best linear approximation to the function at a point is invertible, then with sufficient regularity
Inverse_function_theorem
Fractal composed of triangles
triangle, so the following algorithm will again generate arbitrarily close approximations to it: Start by labeling p1, p2 and p3 as the corners of the Sierpiński
Sierpiński_triangle
Type of image blur produced by a Gaussian function
calculations for each pixel. In practice, when computing a discrete approximation of the Gaussian function, pixels at a distance of more than 3σ have
Gaussian_blur
Style of formal logical argumentation
argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than David
Sequent_calculus
Image noise reducing technique
edges. Bilateral filter Edge detection Edge-preserving smoothing Heat equation Image noise Noise reduction Scale space Total variation denoising Bounded
Anisotropic_diffusion
Technique in natural language processing
by k respectively. The SVD operation, along with this reduction, has the effect of preserving the most important semantic information in the text while
Latent_semantic_analysis
the data is sampled well enough, the resulting graph is a discrete approximation of the underlying manifold. The neighbourhood graph is "unfolded" with
Semidefinite_embedding
Type of sideband modulation
truncation of the spectrum and nonlinear phase distortion from the approximation errors of the practical implementation of the required Hilbert transform
Controlled-envelope single-sideband modulation
Controlled-envelope_single-sideband_modulation
Problem-solving method
perfected, or rationalized, but is nevertheless "good enough" as an approximation or attribute substitution. Where finding an optimal solution is impossible
Heuristic
Geometric algorithm
Diffusion maps is a dimensionality reduction or feature extraction algorithm introduced by Coifman and Lafon which computes a family of embeddings of
Diffusion_map
Matrix decomposition
applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix. The SVD
Singular_value_decomposition
Smooth approximation of one-hot arg max
generalization of the logistic function. The Softmax function is a smooth approximation to the arg max function: the function whose value is the index of a
Softmax_function
Image processing technique
addresses the problem of strong contrast reduction from the scene radiance to the displayable range while preserving the image details and color appearance
Tone_mapping
Atmospheric phenomenon
association between ocular cortical cataracts and UV-B exposure, using crude approximations of exposure and various cataract assessment techniques. A detailed assessment
Ozone_depletion
Distribution of an uncertain quantity
the ball will be found under; the uniform prior is the only one which preserves this invariance. If one accepts this invariance principle then one can
Prior_probability
Signal processing technique
sensitive materials. Low-density parity-check code Noiselet Sparse approximation Sparse coding Verification-based message-passing algorithms in compressed
Compressed_sensing
Typographical mark (\)
characters and to introduce lambda functions (since it is a reasonable approximation in ASCII of the Greek letter lambda, λ). MS-DOS 2.0, released 1983,
Backslash
Artificial neural network node function
be a universal function approximator. This is known as the Universal Approximation Theorem. The identity activation function does not satisfy this property
Activation_function
Computer-aided geometric design
In mathematics, the progressive-iterative approximation method is an iterative method of data fitting with geometric meanings. Given a set of data points
Progressive-iterative approximation method
Progressive-iterative_approximation_method
Moist tobacco product placed under the upper lip
that facilitate the formation of tobacco-specific nitrosamines, while preserving the desired texture and mouthfeel of the snus. The absorption of nicotine
Snus
Spanish variety spoken in Gibraltar
someone is acting with excessive self-importance. Melbil is a Spanish approximation of the pronunciation of the British name Melville, and the expression
Llanito
Problem of determining if a Boolean formula could be made true
formulas, sometimes called CNFSAT. A useful property of Cook's reduction is that it preserves the number of accepting answers. For example, deciding whether
Boolean satisfiability problem
Boolean_satisfiability_problem
Mammoplasty procedure to raise sagging breasts
procedure. Moreover, mastopexy surgery techniques also are applied to reduction mammoplasty, which is the correction of oversized breasts. Psychologically
Mastopexy
priori probability Abductive reasoning Absolute deviation Absolute risk reduction Absorbing Markov chain ABX test Accelerated failure time model Acceptable
List_of_statistics_articles
Statistical method
entropy principles satisfying the Ergodic theorem with mean-preserving and mass-preserving constraints. There is an R package, meboot, that utilizes the
Bootstrapping_(statistics)
Nonparametric test of the null hypothesis
samples, the distribution is tabulated For sample sizes above ~20, approximation using the normal distribution is fairly good. Alternatively, the null
Mann–Whitney_U_test
Mathematical transform
small subset of labeled nodes, theoretically operating as a first order approximation of spectral graph convolutions without computing the graph Laplacian
Graph_Fourier_transform
Ratio of two numbers
multiply 15 by 1⁄3, for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions
Fraction
Form of encryption that allows computation on ciphertexts
method for evaluating polynomial approximations, and is the preferred approach for implementing privacy-preserving machine learning applications. The
Homomorphic_encryption
Head of government of the United Kingdom
executive authorities, but in truth, its merit consists in their singular approximation. The connecting link is the Cabinet ... A Cabinet is a combing committee—a
Prime Minister of the United Kingdom
Prime_Minister_of_the_United_Kingdom
between two phases: drafting and verification. During drafting, a fast approximation model generates a short run of K candidate tokens, typically between
Speculative_decoding
Mathematical result
applications in compressed sensing, manifold learning, dimensionality reduction, graph embedding, and natural language processing. Much of the data stored
Johnson–Lindenstrauss_lemma
Disorder caused by dissolved gases forming bubbles in tissues
decompression sickness which uses the dive profile as input and a mathematical approximation of tissue gas dynamics which omits a number of factors which are known
Decompression_sickness
DC-DC voltage step-down power converter
off-state). The increase in average current makes up for the reduction in voltage, and ideally preserves the power provided to the load. During the off-state
Buck_converter
Comparison of languages
ມະລາບີ M-A-L-A-B-I malabi /māʔ.láː.bìː/, approximation of endonym of the Mlabri people. Thai orthography preserves writing the consonants together, although
Comparison_of_Lao_and_Isan
Italian computer scientist
Giorgio; D'Atri, Alessandro; Protasi, Marco (1980). "Structure preserving reductions among convex optimization problems". Journal of Computer and System
Giorgio_Ausiello
Technique for the generative modeling of a continuous probability distribution
Given a density q {\displaystyle q} , we wish to learn a score function approximation f θ ≈ ∇ ln q {\displaystyle f_{\theta }\approx \nabla \ln q} . This
Diffusion_model
Modification or development of a language
least effort tends to result in phonetic reduction of speech forms. See vowel reduction, cluster reduction, lenition, and elision. After some time, a
Language_change
Standardized means of organizing and storing digital images
A compression algorithm stores either an exact representation or an approximation of the original image in a smaller number of bytes that can be expanded
Image_file_format
Human-caused changes to climate on Earth
adopts the 51-year reference period, 1850–1900 inclusive, assessed as an approximation of pre-industrial levels in AR5 ... Temperatures rose by 0.0 °C–0.2 °C
Climate_change
Slanting line punctuation mark (/)
backslash, the symbol chosen by this transcript as a typographical approximation thereof. For an example of this in practice, see the section on proofreading
Slash_(punctuation)
Greek mathematician and astronomer (c. 240–300)
Philo of Gadara, of whom nothing is known about other than he gave an approximation for π better than Archimedes. In geometry, Sporus seemed to dwell on
Sporus_of_Nicaea
Branch of elementary mathematics
the next digit is less than 5, so that the rounded number is the best approximation of a given precision for the original number. For instance, if the number
Arithmetic
Species of elementary particle
not substantially contribute to the system's behavior, and to zeroth approximation the masses of the lightest quarks can be ignored for most purposes,
Flavour_(particle_physics)
APPROXIMATION PRESERVING-REDUCTION
APPROXIMATION PRESERVING-REDUCTION
Girl/Female
Indian
Dedicate, Presenting
Boy/Male
Muslim
Patient, Tolerant, Forbearing, Preserving
Boy/Male
American, Australian, British, Danish, Dutch, English, French, German, Jamaican, Latin, Swedish, Swiss
Lame; Preserving; Disabled; Limping
Girl/Female
Indian
Dedicate, Presenting
Boy/Male
Tamil
Persevring
Boy/Male
Hindu
Persevring
Boy/Male
British, English
From the Preserving Land
Male
Chinese
preserving depth.
Boy/Male
Indian
Patient, Tolerant, Forbearing, Preserving
Boy/Male
Indian
Patient, Tolerant, Forbearing, Preserving
Girl/Female
Gujarati, Hindu, Indian, Kannada, Marathi, Sanskrit, Sindhi, Tamil, Telugu
Deserving Praises
Boy/Male
English
From the preserving land.
Boy/Male
Muslim
Patient, Tolerant, Forbearing, Preserving
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Deserving; Respectable
Girl/Female
Tamil
Arpita | à®…à®°à¯à®ªà®¿à®¤à®¾
Dedicate, Presenting
Arpita | à®…à®°à¯à®ªà®¿à®¤à®¾
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Preserving; Protecting
Girl/Female
Muslim/Islamic
Patient preservant
Boy/Male
Arabic, Muslim
Preserving Himself from Sin
Girl/Female
Tamil
Arpitha | à®…à®°à¯à®ªà®¿à®¤à®¾
Dedicate, Presenting
Arpitha | à®…à®°à¯à®ªà®¿à®¤à®¾
Boy/Male
Muslim/Islamic
Deserving
APPROXIMATION PRESERVING-REDUCTION
APPROXIMATION PRESERVING-REDUCTION
Girl/Female
Indian, Punjabi, Sikh
Love
Biblical
paternal; that pursues the steps of his father
Boy/Male
Indian, Sikh
Lovable
Girl/Female
Indian
Powerful, Power, Diamond, Darkness
Boy/Male
Arabic, Australian, German, Muslim, Pashtun
The Generous; The Giving; The Chosen
Girl/Female
Hindu, Indian
Humility
Surname or Lastname
English, Scottish, and northern Irish
English, Scottish, and northern Irish : Anglicized form of French Giles. This is believed to be a Huguenot name.
Boy/Male
Hindu, Indian, Telugu
Son
Girl/Female
Greek Latin American
Christian.
Girl/Female
Norse
One of Frigga's ladies in waiting.
APPROXIMATION PRESERVING-REDUCTION
APPROXIMATION PRESERVING-REDUCTION
APPROXIMATION PRESERVING-REDUCTION
APPROXIMATION PRESERVING-REDUCTION
APPROXIMATION PRESERVING-REDUCTION
a.
Meritorious; worthy; as, a deserving person or act.
n.
One who, or that which, approximates.
p. pr. & vb. n.
of Present
p. pr. & vb. n.
of Preserve
n.
A value that is nearly but not exactly correct.
n.
A continual approach or coming nearer to a result; as, to solve an equation by approximation.
a.
Preserving; rescuing.
p. pr. & vb. n.
of Reserve
a.
Serving as a monument; memorial; preserving memory.
a.
Approaching; approximate.
adv.
With approximation; so as to approximate; nearly.
n.
A book or case for preserving dried plants.
a.
Preserving health; prophylactic.
a.
Counteracting, or preserving from, putrefaction; antiseptic.
n.
The act of approximating; a drawing, advancing or being near; approach; also, the result of approximating.
a.
Defending or preserving from disease; preventive.
n.
The transient approximation of the edges of a natural opening; imperforation.
n.
An approach to a correct estimate, calculation, or conception, or to a given quantity, quality, etc.
p. pr. & vb. n.
of Approximate
a.
Resembling, or approximating to, a hemisphere in form.