Parameterized complexity: Difference between revisions

Revision as of 23:42, 16 November 2023

Revision as of 04:02, 24 November 2023

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Under the assumption that [[P versus NP problem|P ≠ NP]], there exist many natural problems that require superpolynomial [[running time]] when complexity is measured in terms of the input size only but that are computable in a time that is polynomial in the input size and exponential or worse in a parameter {{mvar|k}}. Hence, if {{mvar|k}} is fixed at a small value and the growth of the function over {{mvar|k}} is relatively small then such problems can still be considered "tractable" despite their traditional classification as "intractable".

The existence of efficient, exact, and deterministic solving algorithms for [[NP-complete]], or otherwise [[NP-hard]], problems is considered unlikely, if input parameters are not fixed; all known solving algorithms for these problems require time that is [[Exponential time|exponential]] (so in particular superpolynomial) in the total size of the input. However, some problems can be solved by algorithms that are exponential only in the size of a fixed parameter while polynomial in the size of the input. Such an algorithm is called a [[fixed-parameter tractable]] (fpt-FPT) algorithm, because the problem can be solved efficiently (i.e., in polynomial time) for constant values of the fixed parameter.

Problems in which some parameter {{mvar|k}} is fixed are called parameterized problems. A parameterized problem that allows for such an fpt-FPT algorithm is said to be a '''fixed-parameter tractable''' problem and belongs to the class {{sans-serif|FPT}}, and the early name of the theory of parameterized complexity was '''fixed-parameter tractability'''.

Many problems have the following form: given an object {{mvar|x}} and a nonnegative integer {{mvar|k}}, does {{mvar|x}} have some property that depends on {{mvar|k}}? 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 challenging to find an algorithm that is exponential ''only'' in {{mvar|k}}, and not in the input size.