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{{Short description|Branch of computational complexity theory}}

En [[ciencias de la computación]], la '''complejidad parametrizada''' es una rama de la [[teoría de la complejidad computacional]] que se centra en la clasificación de [[problemas computacionales]] de acuerdo a su dificultad con respecto a ''varios'' parámetros de la entrada. La complejidad de un problema se expresa entonces mediante una función en esos parámetros. Esto permite clasificar los problemas [[NP-duros]] en una escala más fina que en la configuración clásica, donde la complejidad de un problema sólo se mide por el número de bits en la entrada. Los primeros aportes sobre complejidad parametrizada fueron realizados por {{harvtxt|Downey|Fellows|1999}}.

In [[computer science]], '''parameterized complexity''' is a branch of [[computational complexity theory]] that focuses on classifying [[computational problems]] according to their inherent difficulty with respect to ''multiple'' parameters of the input or output. The complexity of a problem is then measured as a [[Function (mathematics)|function]] of those parameters. This allows the classification of [[NP-hard]] problems on a finer scale than in the classical setting, where the complexity of a problem is only measured as a function of the number of bits in the input. This appears to have been first demonstrated in {{harvtxt|Gurevich|Stockmeyer|Vishkin|1984}}. The first systematic work on parameterized complexity was done by {{harvtxt|Downey|Fellows|1999}}.

Under the assumption that [[P versus NP problem|P ≠ NP]], there exist many natural problems that require super-polynomial [[running time]] when complexity is measured in terms of the input size only but that are computable in a time that is polynomial in the input size and exponential or worse in a parameter {{mvar|k}}. Hence, if {{mvar|k}} is fixed at a small value and the growth of the function over {{mvar|k}} is relatively small then such problems can still be considered "tractable" despite their traditional classification as "intractable".

Bajo el supuesto de que [[P versus NP problem|P ≠ NP]], existen muchos problemas naturales que requieren un [[tiempo computacional]] superpolinomial cuando la complejidad se mide en términos del tamaño de la entrada solamente, pero que son computables en un tiempo polinomial con respecto al tamaño de la entrada y exponencial o peor en un parámetro ''k''. Por lo tanto, si ''k'' se fija en un valor pequeño y el crecimiento de ''k'' es relativamente pequeño, entonces este tipo de problemas todavía puede considerarse "manejable" a pesar de su clasificación tradicional como "intratable".

The existence of efficient, exact, and deterministic solving algorithms for [[NP-complete]], or otherwise [[NP-hard]], problems is considered unlikely, if input parameters are not fixed; all known solving algorithms for these problems require time that is [[Exponential time|exponential]] (so in particular super-polynomial) in the total size of the input. However, some problems can be solved by algorithms that are exponential only in the size of a fixed parameter while polynomial in the size of the input. Such an algorithm is called a [[fixed-parameter tractable]] (FPT) algorithm, because the problem can be solved efficiently (i.e., in polynomial time) for constant values of the fixed parameter.

La existencia de algoritmos eficientes, exactos y deterministas para solucionar problemas [[NP-completo]], o por otra parte [[NP-duro]], se considera poco probable, si los parámetros de entrada no son fijos; todos los algoritmos conocidos para resolver estos problemas requieren [[tiempo exponencial]] (o al menos superpolinomial) en el tamaño total de la entrada. Sin embargo, algunos problemas pueden ser resueltos por algoritmos que son sólo exponencial en el tamaño de un parámetro fijo y a la vez polinomiales en el tamaño de la entrada. Tales algoritmos son llamados [[fixed-paramater tractable]] (fpt-algorithm), debido a que el problema puede resolverse eficientemente para valores pequeños del parámetro fijo.

ProblemasProblems enin loswhich quesome separameter fije{{mvar|k}} algúnis parámetrofixed kare secalled llamanparameterized problemas parametrizadosproblems. UnA problemaparameterized parametrizadoproblem porthat algúnallows algoritmofor FPTsuch sean diceFPT quealgorithm esis unsaid '''problemato tratablebe de parámetroa fijo''' ('''fixed-parameter tractable''') yproblem perteneceand abelongs lato clasethe <math>class {{sans-serif|FPT</math>}}, deand ahíthe queearly elname primerof nombrethe quetheory recibieraof laparameterized teoríacomplexity de complejidad parametrizada fue '''tratabilidad de parámetro fijo'''was ('''fixed-parameter tractability''').

Many problems have the following form: given an object {{mvar|x}} and a nonnegative integer {{mvar|k}}, does {{mvar|x}} have some property that depends on {{mvar|k}}? For instance, for the [[vertex cover problem]], the parameter can be the number of vertices in the cover. In many applications, for example when modelling error correction, one can assume the parameter to be "small" compared to the total input size. Then it is challenging to find an algorithm that is exponential ''only'' in {{mvar|k}}, and not in the input size.

Muchos problemas tienen la siguiente forma: dado un objeto <math>x</math> y un entero no negativo ''k'', determinar si ''x'' cumple alguna propiedad que depende de ''k''. Por ejemplo, para el problema de [[cobertura de vértices]], el parámetro puede ser el número de vértices en la cubertura. En muchas aplicaciones, por ejemplo, al modelar la corrección de errores, se puede asumir que el parámetro va a ser “pequeño” comparado con el tamaño total de la entrada. Entonces es interesante ver si podemos encontrar un algoritmo que sea exponencial ''sólo'' en ''k'', y no en el tamaño de entrada.

In this way, parameterized complexity can be seen as ''two-dimensional'' complexity theory. This concept is formalized as follows:

De esta manera, la complejidad parametrizada puede verse como un tipo de teoría de la complejidad de ''dos dimensiones''. Este concepto se formaliza de la siguiente forma:

:UnA ''problemaparameterized parametrizadoproblem'' esis una lenguajelanguage <math>L \subseteq \Sigma^* \times \N</math>, dondewhere <math>\Sigma</math> esis una alfabetofinite finitoalphabet. ElThe segundosecond componentecomponent seríais elcalled the ''parámetroparameter'' delof the problemaproblem.

:UnA problemaparameterized parametrizadoproblem <math>{{mvar|L</math>}} esis un''fixed-parameter fpttractable'' siif lathe interrogantequestion “¿"<math>(x, k) \in L</math>?”" puedecan serbe resueltadecided enin running tiempotime <math>f(k) \cdot |x|^{O(1)}</math>, dondewhere <math>{{mvar|f</math>}} esis unaan funciónarbitrary arbitrariafunction quedepending dependeonly sóloon de <math>{{mvar|k</math>}}. La clase deThe complejidadcorresponding [[complexity correspondienteclass]] seis llamacalled '''FPT'''.

:A parameterized problem uses the ''natural parameter'' when its parameter is the size of the solution to the problem.

Por ejemplo, hay un algoritmo que resuelve el problema de cobertura de vértices en tiempo <math>O(kn + 1.274^k)</math>, <ref>{{harvnb|Chen|Kanj|Xia|2006}}</ref> donde <math>n</math> es el número de vértices y <math>k</math> es el tamaño de la cobertura. Esto significa que la cobertura de vértice es fpt tomando como parámetro el tamaño de la solución.

For example, there is an algorithm that solves the vertex cover problem in <math>O(kn + 1.274^k)</math> time,<ref>{{harvnb|Chen|Kanj|Xia|2006}}</ref> where {{mvar|n}} is the number of vertices and {{mvar|k}} is the size of the vertex cover. This means that vertex cover is fixed-parameter tractable with the size of the solution as the parameter (its natural parameter).

==Clases de Complejidad==

== Complexity classes ==

===FPT===

=== FPT ===

Contiene los ''problemas tratables de parámetro fijo'', los cuales pueden ser resueltos en tiempo <math>f(k) \cdot {|x|}^{O(1)}</math> para alguna función computable ''f''. Por lo general, esta función se considera como única exponencial, como <math>2^{O(k)}</math> pero la definición admite funciones que crecen aún más rápido. Esto es esencial para una gran parte de la historia temprana de esta clase. La parte esencial de la definición es excluir funciones de la forma <math>f(n,k)</math> tal como <math>n^k</math>. La clase de parámetro fijo lineal ('''FPL''' por sus siglas en inglés) es la clase de problemas resolubles en tiempo <math>f(k) \cdot |x|</math> para alguna función computable ''f'' [Grohe, 1999]. FPL es una subclase de FPT.

FPT contains the ''fixed parameter tractable'' problems, which are those that can be solved in time <math>f(k) \cdot {|x|}^{O(1)}</math> for some computable function {{mvar|f}}. Typically, this function is thought of as single exponential, such as <math>2^{O(k)}</math>, but the definition admits functions that grow even faster. This is essential for a large part of the early history of this class. The crucial part of the definition is to exclude functions of the form <math>f(n,k)</math>, such as <math>k^n</math>.

The class '''FPL''' (fixed parameter linear) is the class of problems solvable in time <math>f(k) \cdot |x|</math> for some computable function {{mvar|f}}.<ref>{{harvtxt|Grohe|1999}}</ref> FPL is thus a subclass of FPT. An example is the [[Boolean satisfiability]] problem, parameterised by the number of variables. A given formula of size {{mvar|m}} with {{mvar|k}} variables can be checked by brute force in time <math>O(2^km)</math>. A [[vertex cover]] of size {{mvar|k}} in a graph of order {{mvar|n}} can be found in time <math>O(2^kn)</math>, so the vertex cover problem is also in FPL.

Un ejemplo es el problema de [[satisfacibilidad]], parametrizado por el número de variables. Dada una fórmula de tamaño ''m'' con ''k'' variables puede ser verificada mediante fuerza bruta en tiempo <math>O(2^km)</math>. Una [[cobertura de vértices]] de tamaño ''k'' en un grafo de orden ''n'' puede ser encontrada en tiempo <math>O(2^kn)</math>, por tanto este problema está también en FPT.

An example of a problem that is thought not to be in FPT is [[graph coloring]] parameterised by the number of colors. It is known that 3-coloring is [[NP-hard]], and an algorithm for graph {{mvar|k}}-coloring in time <math>f(k)n^{O(1)}</math> for <math>k=3</math> would run in polynomial time in the size of the input. Thus, if graph coloring parameterised by the number of colors were in FPT, then [[P versus NP problem|P&nbsp;=&nbsp;NP]].

Un ejemplo de un problema que no pertenece a esta clase es la [[coloración de un grafo]] parametrizada por el número de colores. Se conoce que el problema de saber si un grafo se puede colorear con a lo sumo 3 colores es [[NP-duro]] y un algoritmo para grafos ''k''-coloreables de tiempo <math>f(k)n^{O(1)}</math> para ''k''=3 corre en tiempo polinomial en el tamaño de la entrada. Por tanto, si este problema es parametrizado en el número de colores dentro de FPT, entonces P=NP.

HayThere variasare alternativasa number of alternative paradefinitions definirof FPT. PorFor ejemploexample, elthe requerimientorunning-time derequirement tiempocan computacionalbe puedereplaced ser remplazado porby <math>f(k) + |x|^{O(1)}</math>. TambiénAlso, una problemaparameterised parametrizadoproblem estáis enin FPT siif esteit tienehas una ciertoso-called kernel. La [[KernelizaciónKernelization]] esis una preprocesopreprocessing técnicotechnique quethat reducereduces la instanciathe original ainstance esteto “kernelits fuerte”"hard kernel", unaa posiblepossibly instanciamuch muchosmaller másinstance pequeñathat queis esequivalent equivalenteto athe laoriginal instanciainstance originalbut perohas tienea unsize tamañothat acotadois porbounded unaby a funciónfunction enin elthe parámetroparameter.

FPT is closed under a parameterised notion of [[Reduction (complexity)|reductions]] called '''''fpt-reductions'''''. Such reductions transform an instance <math>(x,k)</math> of some problem into an equivalent instance <math>(x',k')</math> of another problem (with <math>k' \leq g(k)</math>) and can be computed in time <math>f(k)\cdot p(|x|)</math> where <math>p</math> is a polynomial.

FPT es encerrada dentro de una [[Reduction (complexity)|reducción]] parametrizada llamada '''''fpt-reduction''''', la cual simultáneamente preserva el tamaño de la instancia y el parámetro.

ObviamenteObviously, FPT contienecontains aall todos los problemas depolynomial-time ordencomputable polinomialproblems. AdemásMoreover, contieneit acontains todosall losoptimisation problemasproblems de optimización enin NP quethat tienenallow unan esquema[[Efficient depolynomial-time aproximaciónapproximation polinomial ([[Fullyscheme|efficient polynomial-time approximation scheme (EPTAS)]]).

=== ''W'' hierarchy ===

The '''''W'' hierarchy''' is a collection of computational complexity classes. A parameterized problem is in the class ''W''[''i''], if every instance <math>(x, k)</math> can be transformed (in fpt-time) to a combinatorial circuit that has [[weft (circuit)|weft]] at most ''i'', such that <math>(x, k)\in L</math> if and only if there is a satisfying assignment to the inputs that assigns 1 to exactly ''k'' inputs. The '''weft''' is the largest number of logical units with fan-in greater than two on any path from an input to the output. The total number of logical units on the paths (known as depth) must be limited by a constant that holds for all instances of the problem.

Note that <math>\mathsf{FPT} = W[0]</math> and <math>W[i] \subseteq W[j]</math> for all <math>i\le j</math>. The classes in the ''W'' hierarchy are also closed under fpt-reduction.

===Jerarquía ''W''===

A complete problem for ''W''[''i''] is '''Weighted ''i''-Normalized Satisfiability''':<ref>{{cite journal |last1=Downey |first1=Rod G. |last2=Fellows |first2=Michael R. |title=Fixed-Parameter Tractability and Completeness I: Basic Results |journal=SIAM Journal on Computing |date=August 1995 |volume=24 |issue=4 |pages=873–921 |doi=10.1137/S0097539792228228 |url=https://doi.org/10.1137/S0097539792228228 |language=en |issn=0097-5397|url-access=subscription }}</ref> given a Boolean formula written as an AND of ORs of ANDs of ... of possibly negated variables, with <math>i+1</math> layers of ANDs or ORs (and ''i'' alternations between AND and OR), can it be satisfied by setting exactly ''k'' variables to 1?

Es una colección de clases de complejidad computacional. Un problema parametrzado está en la clase ''W[''i''], si toda instancia <math>(x, k)</math> puede ser transformada (en tiempo fpt) a un camino combinatorio que tenga un weft de a lo sumo ''i'', tal que <math>(x, k)\in L</math> si y solo si existe una asignación satisfactoria para la entrada, la cual asigne ''1'' a lo sumo ''k'' entradas. La altura es el mayor número de unidades lógicas en algún camino desde la entrada hasta la salida. El numero de unidades lógicas acotadas en el camino debe ser limitado por una constante que encierre a todas las instancias del problema.

Notar que FPT&nbsp;=&nbsp;''W''[0] y W[''i''] <math>\subseteq</math> ''W''[''j''] para todo <math>i\le j</math>. Las clases en la jerarquía ''W'' son también encerradas dentro de fpt-reduction.

Muchos problemas computacionales naturales ocupan los niveles más bajos, ''W''[1] y ''W''[2].

Many natural computational problems occupy the lower levels, ''W''[1] and ''W''[2].

====''W''[1]====

==== ''W''[1] ====

Ejemplos de problemas ''W''[1]-completo pueden ser

{{Redirect|W(1)|the mathematical constant|omega constant}}

* Decidir si un grafo dado contiene un [[Clique (graph theory)|clique]] de tamaño ''k''

Examples of ''W''[1]-complete problems include

* Decidir si un grafo dado contiene un [[Independent set (graph theory)|conjunto independiente]] de tamaño ''k''

* deciding if a given graph contains a [[Clique (graph theory)|clique]] of size ''k''

* Decidir si un autómata no determinista reconoce la cadena con ''k'' pasos (problema de “aceptación mínima de un autómata”).

* deciding if a given graph contains an [[Independent set (graph theory)|independent set]] of size ''k''

* deciding if a given nondeterministic single-tape Turing machine accepts within ''k'' steps ("short Turing machine acceptance" problem). This also applies to nondeterministic [[Turing machine|Turing machines]] with ''f''(''k'') tapes and even ''f''(''k'') of ''f''(''k'')-dimensional tapes, but even with this extension, the restriction to ''f''(''k'') tape alphabet size is fixed-parameter tractable. Crucially, the branching of the Turing machine at each step is allowed to depend on ''n'', the size of the input. In this way, the Turing machine may explore ''n''^O(''k'') computation paths.

==== ''W''[2] ====

Examples of ''W''[2]-complete problems include

* deciding if a given graph contains a [[dominating set]] of size ''k''

Ejemplos de problemas ''W''[2]-completos pueden ser

* deciding if a given nondeterministic [[Turing machine equivalents#Multi-tape Turing machines|multi-tape Turing machine]] accepts within ''k'' steps ("short multi-tape Turing machine acceptance" problem). Crucially, the branching is allowed to depend on ''n'' (like the W[1] variant), as is the number of tapes. An alternate ''W''[2]-complete formulation allows only single-tape Turing machines, but the alphabet size may depend on ''n''.

* Decidir si un grafo dado contiene un [[conjunto dominante]] de tamaño ''k''.

* Decidir una [[Turing machine equivalents#Multi-tape Turing machines|máquina de Turing con cinta multi-pista]] reconoce la cadena usando ''k'' pasos (problema de “aceptación mínima de un máquina de Turing con cinta multi-pista”).

==== ''W''[''t''] ====

<math>W[t]</math> can be defined using the family of Weighted Weft-{{mvar|t}}-Depth-{{mvar|d}} SAT problems for <math>d\geq t</math>:

<math>W[t,d]</math> is the class of parameterized problems that fpt-reduce to this problem, and <math>W[t] = \bigcup_{d\geq t} W[t,d]</math>.

Here, '''Weighted Weft-{{mvar|t}}-Depth-{{mvar|d}} SAT''' is the following problem:

Puede ser definido usando la familia de problemas '''Weighted Weft-<math>t</math>-Depth-<math>d</math> SAT''' con <math>d\geq t</math> : <math>W[t,d]</math> es la clase de problemas parametrizados que se fpt-reduce a este problema, y <math>W[t] = \bigcup_{d\geq t} W[t,d]</math>.

* Input: A Boolean formula of depth at most {{mvar|d}} and weft at most {{mvar|t}}, and a number {{mvar|k}}. The ''depth'' is the maximal number of gates on any path from the root to a leaf, and the ''weft'' is the maximal number of gates ''of fan-in at least three'' on any path from the root to a leaf.

El problema '''Weighted Weft-<math>t</math>-Depth-<math>d</math> SAT''' puede ser enunciado de la manera siguiente:

* Question: Does the formula have a satisfying assignment of [[Hamming weight]] exactly {{mvar|k}}?

It can be shown that for <math>t\geq2</math> the problem Weighted {{mvar|t}}-Normalize SAT is complete for <math>W[t]</math> under fpt-reductions.<ref>{{cite journal |last1=Buss |first1=Jonathan F |last2=Islam |first2=Tarique |title=Simplifying the weft hierarchy |journal=[[Theoretical Computer Science (journal)|Theoretical Computer Science]] |year=2006 |volume=351 |number=3 |pages=303–313 |doi=10.1016/j.tcs.2005.10.002|doi-access=free }}</ref>

* Entrada: Una formula boleana de profundidad a lo sumo <math>d</math> y "weft" menor que <math>t</math>, y un número <math>k</math>. La ''profundidad'' es el máximo número de nodos en algún camino desde la raíz a una hoja, y el ''weft'' es el máximo número de nodos de entradas en algún camino de la raíz a una hoja.

Here, '''Weighted {{mvar|t}}-Normalize SAT''' is the following problem:

* Pregunta: ¿Existe una asignación satisfactoria para dicha fórmula en la que ''Hamming weight'' sea menor que <math>k</math>?

* Input: A Boolean formula of depth at most {{mvar|t}} with an AND-gate on top, and a number {{mvar|k}}.

Se puede mostrar que el problema Weighted <math>t</math>-Normalize SAT es completo para <math>W[t]</math> dentro de fpt-reductions.<ref>{{cite journal| author=Buss, Jonathan F, Islam, Tarique| title=Simplifying the weft hierarchy| journal=Theoretical Computer Science| year=2006| volume=351| number=3| pages=303–313| publisher=Elsevier| accessdate=16 April 2014| doi=10.1016/j.tcs.2005.10.002}}</ref> Este problema se puede plantear como:

* Question: Does the formula have a satisfying assignment of Hamming weight exactly {{mvar|k}}?

==== ''W''[''P''] ====

* Entrada: Una fórmula boleana con profundidad a lo sumo <math>t</math> y un AND en el tope, y un número <math>k</math>.

''W''[''P''] is the class of problems that can be decided by a nondeterministic <math>h(k) \cdot {|x|}^{O(1)}</math>-time Turing machine that makes at most <math>O(f(k)\cdot \log n)</math> nondeterministic choices in the computation on <math>(x,k)</math> (a ''k''-restricted Turing machine). {{harvtxt|Flum|Grohe|2006}}

* Pregunta: ¿Existe una asignación satisfactoria para dicha fórmula en la que ''Hamming weight'' sea menor que <math>k</math>?

It is known that FPT is contained in W[P], and the inclusion is believed to be strict. However, resolving this issue would imply a solution to the [[P versus NP]] problem.

====''W''[''P'']====

Other connections to unparameterised computational complexity are that FPT equals ''W''[''P''] if and only if [[circuit satisfiability]] can be decided in time <math>\exp(o(n))m^{O(1)}</math>, or if and only if there is a computable, nondecreasing, unbounded function f such that all languages recognised by a nondeterministic polynomial-time Turing machine using {{tmath|f(n)\log n}} nondeterministic choices are in&nbsp;''P''.

Es la case de problemas que pueden ser decididos en tiempo polinomial mediante un autómata no deterministas en a lo sumo <math>O(f(k)\cdot \log n)</math> para computar <math>(x,k)</math> (a ''k''-restricted Turing-machine).{{harvtxt|Flum|Grohe|2006}}

''W''[''P''] can be loosely thought of as the class of problems where we have a set {{mvar|S}} of {{mvar|n}} items, and we want to find a subset <math>T \subset S</math> of size {{mvar|k}} such that a certain property holds. We can encode a choice as a list of {{mvar|k}} integers, stored in binary. Since the highest any of these numbers can be is {{mvar|n}}, <math>\lceil\log_2 n\rceil</math> bits are needed for each number. Therefore <math>k \cdot \lceil\log_2 n\rceil </math> total bits are needed to encode a choice. Therefore we can select a subset <math>T\subset S</math> with <math>O(k\cdot \log n)</math> nondeterministic choices.

Se sabe que FPT está incluido en ''W''[''P''], y la inclusión es considerada estricta. Sin embargo, resolver este problema debe implicar la solución de la relación [[P contra NP]].

Otras conexiones para la complejidad computacional no parametrizada son que FPT igual ''W''[''P''] si y solo si el [[camino de satisfacibilidad]] puede ser decidido en tiempo <math>\exp(o(n))m^{O(1)}</math>, o si y solo si hay una computable, no decreciente, función f tal que todos los lenguajes reconocido en tiempo polinomial mediante un autómata no determinista usando f(n)log(n) están en ''P''.

=== XP ===

'''XP''' is the class of parameterized problems that can be solved in time <math>n^{f(k)}</math> for some computable function {{mvar|f}}. These problems are called [[slicewise]] polynomial, in the sense that each "slice" of fixed k has a polynomial algorithm, although possibly with a different exponent for each k. Compare this with FPT, which merely allows a different constant prefactor for each value of k. XP contains FPT, and it is known that this containment is strict by diagonalization.

=== para-NP ===

'''XP''' es la clase de problemas parametrizados que pueden ser resueltos en tiempo <math>n^{f(k)}</math> para alguna función <math>f</math> computable.

'''para-NP''' is the class of parameterized problems that can be solved by a [[nondeterministic algorithm]] in time <math>f(k) \cdot |x|^{O(1)}</math> for some computable function {{mvar|f}}. It is known that <math>\textsf{FPT}=\textsf{para-NP}</math> if and only if <math>\textsf{P}=\textsf{NP}</math>.{{sfnp|Flum|Grohe|2006|page=39}}

A problem is '''para-NP-hard''' if it is <math>\textsf{NP}</math>-hard already for a constant value of the parameter. That is, there is a "slice" of fixed {{mvar|k}} that is <math>\textsf{NP}</math>-hard. A parameterized problem that is <math>\textsf{para-NP}</math>-hard cannot belong to the class <math>\textsf{XP}</math>, unless <math>\textsf{P}=\textsf{NP}</math>. A classic example of a <math>\textsf{para-NP}</math>-hard parameterized problem is [[graph coloring]], parameterized by the number {{mvar|k}} of colors, which is already <math>\textsf{NP}</math>-hard for <math>k=3</math> (see [[Graph coloring#Computational complexity]]).

{{Expand section|date=April 2011}}

=== A hierarchy ===

The '''A hierarchy''' is a collection of computational complexity classes similar to the W hierarchy. However, while the W hierarchy is a hierarchy contained in NP, the A hierarchy more closely mimics the [[polynomial-time hierarchy]] from classical complexity. It is known that A[1] = W[1] holds.

== See also ==

* [[Parameterized approximation algorithm]], for [[Optimization problem|optimization problems]] an algorithm running in FPT time might [[Approximation algorithm|approximate]] the solution.

== Notes ==

<references />

== References ==

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|isbn=978-3-319-21274-6

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* {{cite book

|last1 = Downey

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| last2=Grohe | first2=Martin | authorlink2=Martin Grohe

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| title = Parameterized Complexity Theory | year = 2006 | publisher = Springer

|last2 = Fellows

| url = http://www.springer.com/east/home/generic/search/results?SGWID=5-40109-22-141358322-0

|first2 = Michael R.

| isbn = 978-3-540-29952-3 |ref=harv

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|isbn = 978-0-387-94883-6

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* {{cite book

|last1 = Flum

|first1 = Jörg

|authorlink = Jörg Flum

|last2 = Grohe

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|authorlink2 = Martin Grohe

|title = Parameterized Complexity Theory

|year = 2006

|publisher = Springer

|url = https://www.springer.com/east/home/generic/search/results?SGWID=5-40109-22-141358322-0

|isbn = 978-3-540-29952-3

}}

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|last2=Lokshtanov|first2=Daniel

| title=Invitation to Fixed-Parameter Algorithms

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|title=Kernelization: Theory of Parameterized Preprocessing

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*{{cite book

* The Computer Journal. Volume 51, Numbers 1 and 3 (2008). [http://comjnl.oxfordjournals.org/ The Computer Journal]. Special Double Issue on Parameterized Complexity with 15 survey articles, book review, and a Foreword by Guest Editors R. Downey, M. Fellows and M. Langston.

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* Grohe, Martin (1999). Descriptive and Parameterized Complexity, Appeared in Computer Science Logic, 13th International Workshop (CSL'99), Lecture Notes in Computer Science 1683, pp.&nbsp;264 – 273, Springer-Verlag 1999.

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* {{cite book

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|title = Invitation to Fixed-Parameter Algorithms

|publisher = Oxford University Press

|year = 2006

|url = http://www.oup.com/uk/catalogue/?ci=9780198566076

|isbn = 978-0-19-856607-6

|url-status = dead

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* {{cite book

|first = Martin

|last = Grohe

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|chapter = Descriptive and Parameterized Complexity

|title = Computer Science Logic

|publisher = Springer Berlin Heidelberg

|series = Lecture Notes in Computer Science

|volume = 1683

|year = 1999

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* [[The Computer Journal]]. Volume 51, Numbers 1 and 3 (2008). [https://web.archive.org/web/20051125014940/http://comjnl.oxfordjournals.org/ The Computer Journal]. Special Double Issue on Parameterized Complexity with 15 survey articles, book review, and a Foreword by Guest Editors R. Downey, M. Fellows and M. Langston.

== External links ==

Line 127 ⟶ 212:

* [http://www.sprg.uniroma2.it/home/cesati/research/compendium/ Compendium of Parameterized Problems]

[[Category:Parameterized complexity| ]]

[[Category:Computational complexity theory]]

[[Category:Parameterized complexity| ]]

[[en:Uplands]]

[[es:Uplands]]