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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
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
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
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
Branch of computational complexity theory
ISBN 978-3-540-31198-0 Rossmanith, Peter (December 3, 2021). "Parameterized Complexity Theory" (PDF). Parameterized Algorithms (WS 2021/22) (Course lecture slides). RWTH
Parameterized_complexity
Estimate of time taken for running an algorithm
whether the algorithm is allowed to be sub-exponential in the size of the instance, the number of vertices, or the number of edges. In parameterized complexity
Time_complexity
Problem of finding the longest simple path for a given graph
fixed-parameter tractable. The longest path problem, parameterized by clique-width, is hard for the parameterized complexity class W [ 1 ] {\displaystyle W[1]}
Longest_path_problem
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
Technique in numerical linear algebra
endfor The alternating projections algorithm exploits the fact that the low rank approximation problem, parameterized in the image form, is bilinear in
Low-rank_approximation
Class of reinforcement learning algorithms
the policy function πθ is parameterized by a differentiable parameter θ. In policy-based RL, the actor is a parameterized policy function πθ, where θ
Policy_gradient_method
Unsolved problem in computer science
polynomial function on the size of the input to the algorithm. The general class of questions that some algorithm can answer in polynomial time is "P" or "class
P_versus_NP_problem
Reinforcement learning algorithms
RL algorithms. For example, if the critic is estimating the state-value function Vπθ(s), then it can be learned by any value function approximation method
Actor-critic_algorithm
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
Subfield of mathematical optimization
polynomial time and find a solution that is close to optimal parameterized approximation algorithms that run in FPT time and find a solution close to the optimum
Combinatorial_optimization
Task of computing complete subgraphs
best known lower bound is Ω(n), but no matching algorithm is known for the case of k ≥ 3. Parameterized complexity is the complexity-theoretic study of
Clique_problem
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
{\displaystyle \Gamma } . An instance of a parameterized problem consists of (x,k), where k is called the parameter. A parameterized problem Π {\displaystyle \Pi }
Bidimensionality
Probability distribution
algorithm by West (2009) combines Hart's algorithm 5666 with a continued fraction approximation in the tail to provide a fast computation algorithm with
Normal_distribution
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
Probabilistic problem-solving algorithm
experiments or Monte Carlo simulations, are a broad class of computational algorithms based on repeated random sampling for obtaining numerical results, conceptualized
Monte_Carlo_method
Combinatorial optimization graph problem
time approximation schemes (PTAS) were discovered for those problems. While the minimum k-cut problem is W[1]-hard parameterized by k, a parameterized approximation
Minimum_k-cut
Complexity class
NP-complete problems are often addressed by using heuristic methods and approximation algorithms. A decision problem C {\displaystyle \scriptstyle C} is NP-complete
NP-completeness
Matrix decomposition
polar decomposition, and is a common means for implementing low-rank approximation for matrices. Specifically, the singular value decomposition of an m×n
Singular_value_decomposition
Subfield of computational complexity theory
2010. Roditty, Liam; Vassilevska Williams, Virginia (2013). "Fast approximation algorithms for the diameter and radius of sparse graphs". Proceedings of STOC
Fine_grained_complexity
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
Mathematical optimization problem restricted to integers
Koutecký, Martin; Levin, Asaf; Onn, Shmuel (2018). "A parameterized strongly polynomial algorithm for block structured integer programs". In Chatzigiannakis
Integer_programming
Data structure for approximate set membership
hashing techniques were applied. He gave the example of a hyphenation algorithm for a dictionary of 500,000 words, out of which 90% follow simple hyphenation
Bloom_filter
Number of vertices with unambiguous distances
S2CID 231979414 Lokshtanov, Daniel (2010), "Open problems – Parameterized complexity and approximation algorithms: Metric Dimension", in Demaine, Erik D.; Hajiaghayi
Metric dimension (graph theory)
Metric_dimension_(graph_theory)
Method in numerical analysis
continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations, F ( u , λ ) = 0. {\displaystyle F(\mathbf {u}
Numerical_continuation
Probability distribution
variables, each of which has a mean of θ. The gamma distribution can be parameterized in terms of a shape parameter α and an inverse scale parameter β = 1/θ
Gamma_distribution
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
Lossless compression algorithm
are possible, and a dynamic programming algorithm is used to select an optimal one under certain approximations. Prior to LZMA, most encoder models were
LZMA
Algorithm for virtual memory implementation
recently used) approximations and working set algorithms. Since then, some basic assumptions made by the traditional page replacement algorithms were invalidated
Page_replacement_algorithm
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
Calculation of complex statistical distributions
past Integrated nested Laplace approximations Markov chain central limit theorem Metropolis-adjusted Langevin algorithm Robert, Christian; Casella, George
Markov_chain_Monte_Carlo
Curve used in computer graphics and related fields
approximation algorithms have been proposed and used in practice. The rational Bézier curve adds adjustable weights to provide closer approximations to
Bézier_curve
Technique to solve partial differential equations
where u(t,x) denotes the solution, 𝒩[⋅;λ] is a nonlinear operator parameterized by λ, and Ω is a subset of ℝD. This general form of governing equations
Physics-informed neural networks
Physics-informed_neural_networks
American computer scientist
Derandomization and Circuit Lower Bounds". Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz
Russell_Impagliazzo
Annual conference series on algorithms
International Symposium on Parameterized and Exact Computation, founded in 2004 and formerly the International Workshop on Parameterized and Exact Computation
European Symposium on Algorithms
European_Symposium_on_Algorithms
Discrete probability distribution
λ∼Gamma(α,β) denote that λ is distributed according to the gamma density g parameterized in terms of a shape parameter α and an inverse scale parameter β: g(λ∣α
Poisson_distribution
number associated with the graph, used to study the parameterized complexity of graph algorithms. Intuitively, it measures how similar the graph is to
Twin-width
Edges that hit all cycles in a graph
polynomial-time approximation scheme, which generalizes to a weighted version of the problem. A subexponential parameterized algorithm for weighted feedback
Feedback_arc_set
Category of routing problem minimizing total distance and time
Cristina R. Delgado Serna & Joaquín Pacheco Bonrostro applied approximation algorithms to find the best school bus routes in the Spanish province of Burgos
Arc_routing
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
French computer scientist
problems from the points of view of parameterized complexity, fine-grained complexity, approximation algorithms, and regret. Bazgan earned her Ph.D.
Cristina_Bazgan
Straight path on a curved surface or a Riemannian manifold
of different ways to parameterize the shortest path. It is simpler to restrict the set of curves to those that are parameterized "with constant speed"
Geodesic
Measure of similarity between two graphs
often implemented as an A* search algorithm. In addition to exact algorithms, a number of efficient approximation algorithms are also known. Most of them have
Graph_edit_distance
Algorithmic problem in order theory
fixed-parameter tractable when parameterized by the difference q − |C|/3 between the solution quality q found by the parameterized algorithm and the |C|/3 quality
Betweenness_problem
Method in computational phylogenetics
S2CID 14297537. Chen, Jianer; Fan, Jia-Hao; Sze, Sing-Hoi (2015). "Parameterized and approximation algorithms for maximum agreement forest in multifurcating trees"
Tree_rearrangement
is NP-complete. An approximation algorithm with approximation ratio 2 was provided by Frederickson & Ja'Ja' (1981). A parameterized and weighted version
Strong connectivity augmentation
Strong_connectivity_augmentation
P=NP. On the other hand, it was shown that a parameterized 3 / 2 {\displaystyle 3/2} -approximation algorithm with a runtime of 2 O ( k h log h ) n O (
Highway_dimension
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
Computer-aided geometric design
progressive-iterative approximation (PIA) can be divided into interpolation and approximation schemes. In interpolation algorithms, the number of control
Progressive-iterative approximation method
Progressive-iterative_approximation_method
Statistical method of dividing data into equal-sized intervals for analysis
nonparametric estimation algorithms in particular. There are a number of such algorithms such as those based on stochastic approximation or Hermite series estimators
Quantile
Representation of a graph as a path graph "thickened" by some amount
best known approximation ratio of a polynomial time approximation algorithm for pathwidth is O((log n)3/2). For earlier approximation algorithms for pathwidth
Pathwidth
Point set triangulation minimizing total length
polynomial-time approximation algorithms, see Plaisted & Hong (1987) (log-factor approximation) and Levcopoulos & Krznaric (1998) (constant-factor approximation). Cheng
Minimum-weight_triangulation
Multiple-winner electoral system
Kaczmarczyk, Andrzej; Knop, Dusan; Niedermeier, Rolf (2020). "Parameterized Algorithms for Finding a Collective Set of Items". Proceedings of the AAAI
Proportional_approval_voting
Machine learning that combines deep learning and reinforcement learning
variance, making it impractical for use with function approximation in deep RL. Subsequent algorithms have been developed for more stable learning and widely
Deep_reinforcement_learning
Number denoting a graph's closeness to a tree
is commonly used as a parameter in the parameterized complexity analysis of graph algorithms. Many algorithms that are NP-hard for general graphs, become
Treewidth
Any planar graph can be subdivided by removing a few vertices
location, algorithms for polygon triangulation, shortest paths, and the construction of nearest neighbor graphs, and approximation algorithms for the maximum
Planar_separator_theorem
Set of hypergraph nodes to which every hyperedge is connected
d-hitting set permits a d-approximation algorithm. Assuming the unique games conjecture, this is the best constant-factor algorithm that is possible and otherwise
Vertex_cover_in_hypergraphs
Turkish mathematician
are as follows: Onus, M. ve Richa, A.W.(2016). "Parameterized maximum and average degree approximation in topic based publish subscribe overlay network
Melih_Onuş
Graph theory term
Shaowei (2016), "Almost induced matching: linear kernels and parameterized algorithms", in Heggernes, Pinar (ed.), Graph-Theoretic Concepts in Computer
Induced_matching
Function of four real variables that defines how light is reflected at an opaque surface
\omega _{\text{i}}} . Each direction ω {\displaystyle \omega } is itself parameterized by azimuth angle ϕ {\displaystyle \phi } and zenith angle θ {\displaystyle
Bidirectional reflectance distribution function
Bidirectional_reflectance_distribution_function
Problem in computer science
can only be achieved if P = NP. An approximation within a factor of 2 can be achieved with this simple algorithm, however: Output the solution in which
MAX-3SAT
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
Numerical optimization algorithm
towards higher expected fitness. The general procedure is as follows: the parameterized search distribution is used to produce a batch of search points, and
Natural_evolution_strategy
Measure of similarity between curves
discrete Fréchet distance, also called the coupling distance, is an approximation of the Fréchet metric for polygonal curves, defined by Eiter and Mannila
Fréchet_distance
Unit hypercube of variable dimension whose corners have been perturbed
basis-exchange pivoting algorithms and also for interior-point algorithms. The Klee–Minty cube was originally specified with a parameterized system of linear
Klee–Minty_cube
Graph drawing with vertices in horizontal layers
(1990), "Approximation algorithms for the maximum acyclic subgraph problem", Proceedings of the 1st ACM-SIAM Symposium on Discrete Algorithms (SODA'90)
Layered_graph_drawing
Machine Approximation algorithm Simulated annealing Ant colony optimization algorithms Game semantics Generalized game Multiple-agent system Parameterized complexity
List of computability and complexity topics
List_of_computability_and_complexity_topics
Branch of machine learning
is the number of hidden layers plus one (as the output layer is also parameterized). For recurrent neural networks, in which a signal may propagate through
Deep_learning
Machine learning framework
has a respective local operator W t {\displaystyle W_{t}} (usually parameterized by a pointwise neural network), a kernel integral operator K t {\displaystyle
Neural_operators
Numerical invariant of graphs
time algorithm based on the planarity of the excluded minors for tree-depth was given earlier by Bodlaender et al. (1998). For improved parameterized algorithms
Tree-depth
Technique in mathematical modeling
the application of reduced basis methods to parameterized partial differential equations. All algorithms in pyMOR are formulated in terms of abstract
Model_order_reduction
Evolutionary algorithm
quadratic approximation. The update equations for mean and covariance matrix maximize a likelihood while resembling an expectation–maximization algorithm. The
CMA-ES
Probability distribution and special case of gamma distribution
or the chi-squared approximation for a small sample size. Similarly, in analyses of contingency tables, the chi-squared approximation will be poor for a
Chi-squared_distribution
In Bayesian probability theory
to statistical problems such as the Laplace approximation, Gibbs/Metropolis sampling, or the EM algorithm. It is also possible to apply the above considerations
Marginal_likelihood
Maximum number of colors in a greedy graph coloring
approximate the Grundy number to within an approximation ratio better than c. There is an exact exponential time algorithm for the Grundy number that runs in
Grundy_number
Lower bound on the log-likelihood of some observed data
special case when the approximate posterior and the prior are both parameterized as Gaussians, since then the KL term (though it is itself an expectation)
Evidence_lower_bound
Algorithm that estimates unknowns from a series of measurements over time
As such, it is a common sensor fusion and data fusion algorithm. Noisy sensor data, approximations in the equations that describe the system evolution,
Kalman_filter
Mathematical approach to quantum physics
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Type of computational problem
problem admits an r-approximation algorithm, then the conflict-free geometric cover problem admits a similar approximation algorithm in FPT time. Vazirani
Covering_problems
Computer simulations to discover and understand chemical properties
computational cost, force fields employ numerical approximations such as shifted cutoff radii, reaction field algorithms, particle mesh Ewald summation, or the newer
Molecular_dynamics
Dominating set that induces a connected subgraph
(1998), "2-approximation algorithm for finding a spanning tree with maximum number of leaves", Proc. 6th European Symposium on Algorithms (ESA'98), Lecture
Connected_dominating_set
Generalization of the concept of parallel lines
a second curve that can be parameterized by its unit normal, d→(n→), where the normal of d→(n→)=n→ (this parameterization by normal exists for curves
Parallel_curve
Probabilistic classification algorithm
observations in each group), rather than the expensive iterative approximation algorithms required by most other models. Despite the use of Bayes' theorem
Naive_Bayes_classifier
Subdivision of space into cells
Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a
Mesh_generation
Property in graph theory
, the running time of this algorithm is 2 O ( c 2 ) n {\textstyle 2^{O(c^{2})}n} . An alternative parameterized algorithm, more suitable for graphs in
Cutwidth
Branch of chemistry
advances in DFT aim to reduce this complexity through various approximations and algorithmic improvements. There have also been significant speed improvements
Computational_chemistry
Sub-field of computer science
may be non-manifold. Since surfaces are not finite, discrete digital approximations are used. Polygonal meshes (and to a lesser extent subdivision surfaces)
Computer graphics (computer science)
Computer_graphics_(computer_science)
Technique for dimensionality reduction
objects are modeled by distant points with high probability. The t-SNE algorithm comprises two main stages. First, t-SNE constructs a probability distribution
T-distributed stochastic neighbor embedding
T-distributed_stochastic_neighbor_embedding
an efficient algorithm for the egalitarian Monroe rule. Lu and Boutilier presented a polytime 0.63-factor approximation greedy algorithm for the optimal
Fully proportional representation
Fully_proportional_representation
Deep learning generative model to encode data representation
chosen parameterized probability distribution pθ(x)=p(x|θ). This distribution is usually chosen to be a Gaussian N(x|μ,σ) which is parameterized by μ and
Variational_autoencoder
Area of computer graphics
rhythm, tempo, emotional and dynamics handling without complex approximation algorithms. The training database is not needed to be labeled since there
Computer_facial_animation
On linear-time algorithms for graph logic
construction proving that it has bounded clique-width, but later approximation algorithms for clique-width removed this requirement. Courcelle's theorem
Courcelle's_theorem
Classification of Artificial Neural Networks (ANNs)
layers are Input, hidden pattern, hidden summation, and output. In the PNN algorithm, the parent probability distribution function (PDF) of each class is approximated
Types of artificial neural networks
Types_of_artificial_neural_networks
the parameterized complexity of biclique cover and partition", in Guo, Jiong; Hermelin, Danny (eds.), 11th International Symposium on Parameterized and
Graham–Pollak_theorem
Graph with at most one crossing per edge
Michael J.; Cabello, Sergio; Eppstein, David (2013), "Parameterized complexity of 1-planarity", Algorithms and Data Structures Symposium (WADS 2013), vol. 22
1-planar_graph
Solving multiple machine learning tasks at the same time
^{\top }} , the task matrix A † {\displaystyle A^{\dagger }} can be parameterized as a function of M {\displaystyle M} : A † ( M ) = ϵ M U + ϵ B ( M −
Multi-task_learning
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PARAMETERIZED APPROXIMATION-ALGORITHM
PARAMETERIZED APPROXIMATION-ALGORITHM
PARAMETERIZED APPROXIMATION-ALGORITHM
PARAMETERIZED APPROXIMATION-ALGORITHM
PARAMETERIZED APPROXIMATION-ALGORITHM
PARAMETERIZED APPROXIMATION-ALGORITHM
PARAMETERIZED APPROXIMATION-ALGORITHM
PARAMETERIZED APPROXIMATION-ALGORITHM
PARAMETERIZED APPROXIMATION-ALGORITHM
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