Parameterized complexity: Difference between revisions

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: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 &ldquo;<math>(x, k) \in L</math>?&rdquo; 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> the size of the vertex cover. This proves that vertex cover is fixed-parameter tractable with respect to this parameter.

[[Category:Computational complexity theory]]