Parameterized complexity: Difference between revisions

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:A parameterized problem {{mvar|L}} is ''fixed-parameter tractable'' if the question "<math>(x, k) \in L</math>?" can be decided in running time <math>f(k) \cdot |x|^{O(1)}</math>, where {{mvar|f}} is an arbitrary function depending only on {{mvar|k}}. The corresponding [[complexity class]] is called '''FPT'''.

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

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).

== Complexity classes ==

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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.

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?

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