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Class of algorithms that find approximate solutions to optimization problems
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems
Approximation_algorithm
Approximation for the travelling salesman problem
(they are symmetric and obey the triangle inequality). It is an approximation algorithm that guarantees that its solutions will be within a factor of 3/2
Christofides_algorithm
Mathematical and computational problem
produced with sophisticated algorithms. In addition, many approximation algorithms exist. For example, the first fit algorithm provides a fast but often
Bin_packing_problem
Mathematical method that minimizes maximum error
A minimax approximation algorithm (or L∞ approximation or uniform approximation) is a method to find an approximation of a mathematical function that
Minimax approximation algorithm
Minimax_approximation_algorithm
Task of computing complete subgraphs
maximum. Although the approximation ratio of this algorithm is weak, it is the best known to date. The results on hardness of approximation described below
Clique_problem
Sequence of locally optimal choices
matching pursuit is an example of a greedy algorithm applied on signal approximation. A greedy algorithm finds the optimal solution to Malfatti's problem
Greedy_algorithm
Complexity class of approximable problems
polynomial-time approximation algorithms with approximation ratio bounded by a constant (or constant-factor approximation algorithms for short). In simple
APX
Optimization algorithm
Philip Wolfe in 1956. In each iteration, the Frank–Wolfe algorithm considers a linear approximation of the objective function, and moves towards a minimizer
Frank–Wolfe_algorithm
NP-hard problem in combinatorial optimization
matter of fact, the term "algorithm" was not commonly extended to approximation algorithms until later; the Christofides algorithm was initially referred
Travelling_salesman_problem
Unrelated vertices in graphs
trivial algorithm attains a (d − 1)-approximation algorithm for the maximum independent set. In fact, it is possible to get much better approximation ratios:
Independent set (graph theory)
Independent_set_(graph_theory)
Unsolved problem in computational complexity theory
of hardness of approximation. If the UGC is true and P ≠ NP, then for many important problems there not only does not exist an algorithm that always produces
Unique_games_conjecture
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 time
Parameterized approximation algorithm
Parameterized_approximation_algorithm
Problem in graph theory
approximation algorithm achieves an approximation ratio strictly less than one. There is a simple randomized 0.5-approximation algorithm: for each vertex
Maximum_cut
Family of iterative methods
only estimated via noisy observations. In a nutshell, stochastic approximation algorithms deal with a function of the form f ( θ ) = E ξ [ F ( θ , ξ )
Stochastic_approximation
Problem in combinatorial optimization
time algorithm using dynamic programming. There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time algorithm as a
Knapsack_problem
Classification of algorithm
A galactic algorithm is an algorithm with state-of-the-art theoretical (asymptotic) performance, but which is not used due to practical constraints. Typical
Galactic_algorithm
Type of approximation algorithm
computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems
Polynomial-time approximation scheme
Polynomial-time_approximation_scheme
Something roughly the same as something else
An approximation is anything that is intentionally similar but not exactly equal to something else. The word approximation is derived from Latin approximatus
Approximation
Numerical approximation algorithm
quasi-Newton methods like BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called convergent
Iterative_method
Combinatorial optimization problem
any algorithm for the knapsack problem into an approximation algorithm for the GAP. Using any α {\displaystyle \alpha } -approximation algorithm ALG for
Generalized assignment problem
Generalized_assignment_problem
Subfield of convex optimization
important tools for developing approximation algorithms for NP-hard maximization problems. The first approximation algorithm based on an SDP is due to Michel
Semidefinite_programming
Approximation algorithm for the n-body problem
is an approximation algorithm for performing an n-body simulation. It is notable for having order O(n log n) compared to a direct-sum algorithm which
Barnes–Hut_simulation
NP-complete problem in computer science
the runtime is O(n) and the approximation ratio is at most 3/2 ("approximation ratio" means the larger sum in the algorithm output, divided by the larger
Partition_problem
Problem in computational complexity theory
state-of-the-art algorithm is due to Avidor, Berkovitch and Zwick, and its approximation ratio is 0.7968. They also give another algorithm whose approximation ratio
Maximum satisfiability problem
Maximum_satisfiability_problem
Estimate of time taken for running an algorithm
know quasi-polynomial time algorithms, but no polynomial time algorithm is known. Such problems arise in approximation algorithms; a famous example is the
Time_complexity
Classical problem in combinatorics
indeed gives a factor- log n {\displaystyle \scriptstyle \log n} approximation algorithm for the minimum set cover problem. See setcover for a detailed
Set_cover_problem
Curve simplification algorithm
from the line segment is greater than ε from the approximation then that point must be kept. The algorithm recursively calls itself with the first point
Ramer–Douglas–Peucker algorithm
Ramer–Douglas–Peucker_algorithm
Subset of a graph's vertices, including at least one endpoint of every edge
several simple 2-factor approximations. It is a typical example of an NP-hard optimization problem that has an approximation algorithm. Its decision version
Vertex_cover
Problem of finding the longest simple path for a given graph
understanding its approximation hardness". The best polynomial time approximation algorithm known for this case achieves only a very weak approximation ratio, n
Longest_path_problem
Type of analog-to-digital converter
The algorithm starts with an initial approximation of x0 = 0 and during each iteration i produces the following approximation: ith approximation: xi =
Successive-approximation_ADC
Problem in computer science
operations research. It is a problem that is widely taught in approximation algorithms. As input you are given several sets and a number k {\displaystyle
Maximum_coverage_problem
Subfield of mathematical optimization
tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.
Combinatorial_optimization
Decision problem in computer science
where r is a number in (0,1) called the approximation ratio. The following very simple algorithm has an approximation ratio of 1/2: Order the inputs by descending
Subset_sum_problem
Numerical optimization algorithm
This approximation is based on the information matrix equality and therefore only valid while maximizing a likelihood function. The BHHH algorithm is named
Berndt–Hall–Hall–Hausman algorithm
Berndt–Hall–Hall–Hausman_algorithm
Varying methods used to calculate pi
Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning
Approximations_of_pi
2D geometric minimization problem
polynomial-time approximation algorithm with a ratio smaller than 3 / 2 {\displaystyle 3/2} unless P = N P {\displaystyle P=NP} . However, the best approximation ratio
Strip_packing_problem
In computer science, hardness of approximation is a field that studies the algorithmic complexity of finding near-optimal solutions to optimization problems
Hardness_of_approximation
Shape that blocks all lines of sight
provide several linear-time approximation algorithms for the shortest opaque set for convex polygons, with better approximation ratios than two: For general
Opaque_set
Algorithm for the travelling salesman problem
The nearest neighbour algorithm was one of the first algorithms used to solve the travelling salesman problem approximately. In that problem, the salesman
Nearest_neighbour_algorithm
Subset of a graph's nodes such that all other nodes link to at least one
efficient algorithm that can compute γ(G) for all graphs G. However, there are efficient approximation algorithms, as well as efficient exact algorithms for
Dominating_set
Computational complexity class
study approximation algorithms. In particular, a quasi-polynomial-time approximation scheme (QPTAS) is a variant of a polynomial-time approximation scheme
Quasi-polynomial_time
Variant of the traveling salesman problem
preserve the quality of approximations to that solution. If the graph is a metric space then there is an efficient approximation algorithm that finds a Hamiltonian
Bottleneck traveling salesman problem
Bottleneck_traveling_salesman_problem
Mathematical concept
The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation
Approximation_error
This is a list of algorithm general topics. Analysis of algorithms Ant colony algorithm Approximation algorithm Best and worst cases Big O notation Combinatorial
List of algorithm general topics
List_of_algorithm_general_topics
Sequence of operations for a task
While many algorithms reach an exact solution, approximation algorithms seek an approximation that is close to the true solution. Such algorithms have practical
Algorithm
Algorithm in data mining
an approximation algorithm for the NP-hard k-means problem—a way of avoiding the sometimes poor clusterings found by the standard k-means algorithm. It
K-means++
Combinatorial optimization problem
reviewed by Duan and Pettie (see Table II). Their work proposes an approximation algorithm for the assignment problem (and the more general maximum-weight
Assignment_problem
Algorithm for finding zeros of functions
Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function
Newton's_method
Optimization problem in computer science
character recognition Statistical classification – see k-nearest neighbor algorithm Computer vision – for point cloud registration Computational geometry
Nearest_neighbor_search
Linear programming algorithm
Karmarkar's algorithm is an algorithm introduced by Narendra Karmarkar in 1984 for solving linear programming problems. It was the first reasonably efficient
Karmarkar's_algorithm
Quantum algorithm for integer factorization
Shor's algorithm is a quantum algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor
Shor's_algorithm
Algorithm used to solve non-linear least squares problems
In mathematics and computing, the Levenberg–Marquardt algorithm (LMA or just LM), also known as the damped least-squares (DLS) method, is used to solve
Levenberg–Marquardt_algorithm
Technique in numerical linear algebra
In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization
Low-rank_approximation
Population-based search algorithm
computer science and operations research, the bees algorithm is a population-based search algorithm which was developed by Pham, Ghanbarzadeh et al. in
Bees_algorithm
Algorithm to approximate functions
Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934, is an iterative algorithm used to find simple approximations to
Remez_algorithm
Computational geometry and optimization concept
In computational geometry and approximation algorithms, a coreset is a small, possibly weighted subset of an input point set that approximately preserves
Coreset
Method to solve optimization problems
relaxation of a combinatorial problem and are important in the study of approximation algorithms. For example, the LP relaxations of the set packing problem, the
Linear_programming
Algorithm for linear programming
optimization, Dantzig's simplex algorithm (or simplex method) is an algorithm for linear programming. The name of the algorithm is derived from the concept
Simplex_algorithm
Mathematical problem
load-balancing among multiple servers. An approximation algorithm for splitting a necklace can be derived from an algorithm for consensus halving. Combinatorial
Necklace_splitting_problem
Optimization algorithm
an optimization algorithm in the collection of quasi-Newton methods that approximates the Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) using a limited
Limited-memory_BFGS
Mathematical problem
unlikely that any approximation ratio better than some fixed constant can be achieved by a polynomial time approximation algorithm. Ghosh (1987) showed
Art_gallery_problem
Sequential model-based optimization of expensive black-box functions
648–657. Knowles, Joshua (2006). "ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems"
Bayesian_optimization
Optimization by removing non-optimal solutions to subproblems
an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists
Branch_and_bound
Mathematical optimization problem restricted to integers
Branch and bound algorithms have a number of advantages over algorithms that only use cutting planes. One advantage is that the algorithms can be terminated
Integer_programming
Set-to-real map with diminishing returns
property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical
Submodular_set_function
In computer science, k-approximation of k-hitting set is an approximation algorithm for weighted hitting set. The input is a collection S of subsets of
K-approximation of k-hitting set
K-approximation_of_k-hitting_set
Algorithm to compute the maximum flow in a flow network
In computer science, the Edmonds–Karp algorithm is an implementation of the Ford–Fulkerson method for computing the maximum flow in a flow network in
Edmonds–Karp_algorithm
Optimization method
refined by B k {\displaystyle B_{k}} , the approximation to the Hessian. The first step of the algorithm is carried out using the inverse of the matrix
Broyden–Fletcher–Goldfarb–Shanno algorithm
Broyden–Fletcher–Goldfarb–Shanno_algorithm
Vertices whose removal breaks all cycles
existence of an approximation preserving L-reduction from the vertex cover problem to it; Existing constant-factor approximation algorithms. The best known
Feedback_vertex_set
Optimization algorithm
stochastic approximation (SPSA) is an algorithmic method for optimizing systems with multiple unknown parameters. It is a type of stochastic approximation algorithm
Simultaneous perturbation stochastic approximation
Simultaneous_perturbation_stochastic_approximation
Study of mathematical algorithms for optimization problems
perturbation stochastic approximation (SPSA) method for stochastic optimization; uses random (efficient) gradient approximation. Methods that evaluate
Mathematical_optimization
Overview of and topical guide to algorithms
candidate solutions Randomized algorithm — algorithm using randomness as part of its logic Approximation algorithm — algorithm that finds near-optimal solutions
Outline_of_algorithms
Class of problems in computer science
better approximation factors. The approximation ratio of the first such algorithm is asymptotically 2 when k is large, but when k=2 the algorithm achieves
Interval_scheduling
Probabilistic graphical representation of causal relationships
probabilities. The bounded variance algorithm developed by Dagum and Luby was the first provable fast approximation algorithm to efficiently approximate probabilistic
Bayesian_network
Root-finding algorithm
compute an approximation of the base-2 exponential Refine the approximation using a single iteration of Newton's method. Since this algorithm relies heavily
Fast_inverse_square_root
Problem of determining if a Boolean formula could be made true
efficient approximation algorithms, but is NP-hard to solve exactly. Worse still, it is APX-complete, meaning there is no polynomial-time approximation scheme
Boolean satisfiability problem
Boolean_satisfiability_problem
American computer scientist and educator
their work on approximation algorithms for the sparsest cut problem. He was named an ACM Fellow in 2013 for contributions to algorithms for graph partitioning
Satish_B._Rao
Computer science concept
addition to its dynamic programming algorithm, Knuth proposed two heuristics (or rules) to produce nearly (approximation of) optimal binary search trees.
Optimal_binary_search_tree
Term in mathematical optimization
the trust-region approximation, so shrink the trust region (increase λ) and try again. Note that the Levenberg-Marquardt algorithm does not have an explicit
Trust_region
Logic problem, AND of pairwise ORs
assignment that maximizes the number of satisfied constraints, has an approximation algorithm whose optimality depends on the unique games conjecture, and another
2-satisfiability
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
MAXEkSAT
Partition of a graph's nodes into cliques
there can be no polynomial time approximation algorithm for any ε > 0 that, on n-vertex graphs, achieves an approximation ratio better than n1 − ε. In graphs
Clique_cover
Well-spaced set of points in a metric space
well-spaced they are. These sets have applications in coding theory, approximation algorithms, and the theory of quasicrystals. If (M, d) is a metric space,
Delone_set
Algorithms for calculating square roots
computed to some finite precision: these algorithms typically construct a series of increasingly accurate approximations. Most square root computation methods
Square_root_algorithms
Algorithm for computing the maximal flow of a network
Dinic's algorithm or Dinitz's algorithm is a strongly polynomial algorithm for computing the maximum flow in a flow network, conceived in 1970 by Israeli
Dinic's_algorithm
Combinatorial optimization graph problem
sets. Several approximation algorithms exist with an approximation of 2 − 2 k . {\displaystyle 2-{\tfrac {2}{k}}.} A simple greedy algorithm that achieves
Minimum_k-cut
Algorithm in computer science
science and operations research, the artificial bee colony algorithm (ABC) is an optimization algorithm based on the intelligent foraging behaviour of honey
Artificial bee colony algorithm
Artificial_bee_colony_algorithm
Optimization algorithm
unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function. The idea is to
Gradient_descent
Metaheuristic proposed by Xin-She Yang
firefly algorithm is a metaheuristic proposed by Xin-She Yang and inspired by the flashing behavior of fireflies. In pseudocode the algorithm can be stated
Firefly_algorithm
Class of algorithms operating on data streams
required to take action as soon as each point arrives. If the algorithm is an approximation algorithm then the accuracy of the answer is another key factor.
Streaming_algorithm
Line-drawing algorithm
Bresenham's line algorithm is a line drawing algorithm that determines the points of an n-dimensional raster that should be selected in order to form
Bresenham's_line_algorithm
Combinatorial optimization problem
issue by trying all values of k. A simple greedy approximation algorithm that achieves an approximation factor of 2 builds C {\displaystyle {\mathcal {C}}}
Metric_k-center
On short connecting nets with added points
solution can be found by using a polynomial-time algorithm. However, there is a polynomial-time approximation scheme (PTAS) for Euclidean Steiner trees, i
Steiner_tree_problem
A fully polynomial-time approximation scheme (FPTAS) is an algorithm for finding approximate solutions to function problems, especially optimization problems
Fully polynomial-time approximation scheme
Fully_polynomial-time_approximation_scheme
Iterative method for minimizing convex functions
1972, an approximation algorithm for real convex minimization was studied by Arkadi Nemirovski and David B. Yudin (Judin). As an algorithm for solving
Ellipsoid_method
Graph theory problem
bipartite graphs only. Other algorithms are reviewed by Duan and Pettie. Their work also proposes an approximation algorithm for general graphs that runs
Maximum-weight_matching
Optimization algorithm
optimal solution or a close approximation). At the other extreme, bubble sort can be viewed as a hill climbing algorithm (every adjacent element exchange
Hill_climbing
Edges that hit all cycles in a graph
that have a polynomial time approximation algorithm that achieves a constant approximation ratio. Although such approximations are not known for the feedback
Feedback_arc_set
String metric for measuring edit distance
reconstruction, distance-sensitive algorithms, bit-parallel methods, Levenshtein automata, approximation algorithms, and conditional lower bounds on exact
Levenshtein_distance
Fast method for calculating the digits of π
Bailey–Borwein–Plouffe formula Borwein's algorithm Approximations of π Chudnovsky, David; Chudnovsky, Gregory (1988), Approximation and complex multiplication according
Chudnovsky_algorithm
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APPROXIMATION ALGORITHM
APPROXIMATION ALGORITHM
APPROXIMATION ALGORITHM
APPROXIMATION ALGORITHM
APPROXIMATION ALGORITHM
APPROXIMATION ALGORITHM
APPROXIMATION ALGORITHM
APPROXIMATION ALGORITHM
APPROXIMATION ALGORITHM
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