Parameterized complexity

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In computer science, parameterized complexity is a measure of complexity of problems with multiple inputs. It is based on the fact that several such NP-hard are tractable when one of their input is fixed.

The existence of efficient, exact, and deterministic solving algorithms for NP-complete problems problems is considered unlikely, if inputs are not fixed; all known solving algorithma for these problems require time that is exponential in the total size of the inputs. However, some problems can be solved by algorithms that are exponential in the size of one input and polynomial in the size of the other inputs. Such an algorithm is called a fixed-parameter algorithm, because the problem can be solved efficiently by fixing the "troublesome" input at any one value. A problem that allows for such an algorithm is called fixed-parameter tractable, and the early name of the theory of parameterized complexity was fixed-parameter tractability.

Many problems have the following 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 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 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:

A parameterized problem is a language <math>L \subseteq \Sigma^* \times \N</math>, where <math>\Sigma</math> is a finite alphabet. The second component is called the parameter of the problem.

A parameterized problem <math>L</math> 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 <math>f</math> is an arbitrary function depending only on <math>k</math>. The corresponding complexity class is called FPT.

For example, there is an algorithm which solves the vertex cover problem in <math>O(kn + 1.29^k)</math> time, where <math>n</math> is the number of vertices and <math>k</math> is the size of the vertex cover. This proves that vertex cover is fixed-parameter tractable with respect to this parameter.

References

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| first=Rod | last=Downey | coauthors=M. Fellows | title=Parameterized complexity | publisher=Springer | year=1997 | url=http://www.springer.com/sgw/cda/frontpage/0,11855,5-0-22-1519914-0,00.html?referer=www.springer.de%2Fcgi-bin%2Fsearch_book.pl%3Fisbn%3D0-387-94883-X }} ISBN 0-387-94883-X