Living Reference Work Entry

Encyclopedia of Algorithms

pp 1-4

Date: Latest Version

Kernelization, Partially Polynomial Kernels

  • Christian KomusiewiczAffiliated withInstitut für Softwaretechnik und Theoretische Informatik Email author 

Keywords

NP-hard problems Fixed-parameter algorithms Data reduction Kernelization

Years and Authors of Summarized Original Work

2011; Betzler, Guo, Komusiewicz, Niedermeier

2013; Basavaraju, Francis, Ramanujan, Saurabh

2014; Betzler, Bredereck, Niedermeier

Problem Definition

In parameterized complexity, each instance (I, k) of a problem comes with an additional parameter k which describes structural properties of the instance, for example, the maximum degree of an input graph. A problem is called fixed-parameter tractable if it can be solved in f(k) ⋅poly(n) time, that is, the super-polynomial part of the running time depends only on k. Consequently, instances of the problem can be solved efficiently if k is small.

One way to show fixed-parameter tractability of a problem is the design of a polynomial-time data reduction algorithm that reduces any input instance (I, k) to one whose size is bounded in k. This idea is captured by the notion of kernelization.

Definition 1.

Let (I, k) be an instance of a parameterized problem P, where I ∈ Σ ∗  denotes the input instance and ...

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