Parameterized complexity: Difference between revisions

Revision as of 00:12, 18 April 2004

Revision as of 19:42, 3 March 2005

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In [[computer science]], '''fixed-parameter algorithms''' are an approach to attacking [[NP-hard]] problems. When trying to solve these problems exactly and deterministically, one has to deal with exponential running times; [[computational complexity theory]] indicates this is [[Complexity classes P and NP|inevitable]]. Parameterized complexity theory accepts these exponential running times, but claims that not all “intractable” algorithms are equal, and some might even be feasible for practical applications.

The main idea is to consider ''parameters''. Many problems have the following general form: given an object <math>x</math> and a nonnegative integer <math>k</math>, does <math>x</math> have some property that depends on <math>k</math>? For instance, for the [[Vertexvertex Covercover]] 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 &ldquo;small&rdquo; compared to the total input size. Then it is interesting to see whether we can find an algorithm which is exponential ''only'' in <math>k</math>, and not in the input size.

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