Parameterized complexity: Difference between revisions

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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 is considered unlikely, if inputs are not fixed; all known solving algorithmaalgorithms for these problems require time that is [[Exponential time|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. <!-- is this correct? it somehow implies that FPT=constant input --> 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.