# Encyclopedia of Algorithms

Living Edition
| Editors: Ming-Yang Kao

# Kernelization, Partially Polynomial Kernels

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27848-8_530-1

## 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...

### Keywords

NP-hard problems Fixed-parameter algorithms Data reduction Kernelization
This is a preview of subscription content, log in to check access

1. 1.
Basavaraju M, Francis MC, Ramanujan MS, Saurabh S (2013) Partially polynomial kernels for set cover and test cover. In: FSTTCS ’13, Guwahati. Schloss Dagstuhl – Leibniz-Zentrum fuer Informatik, LIPIcs, vol 24, pp 67–78Google Scholar
2. 2.
Betzler N, Guo J, Komusiewicz C, Niedermeier R (2011) Average parameterization and partial kernelization for computing medians. J Comput Syst Sci 77(4):774–789
3. 3.
Betzler N, Bredereck R, Niedermeier R (2014) Theoretical and empirical evaluation of data reduction for exact Kemeny rank aggregation. Auton Agents Multi-Agent Syst 28(5):721–748
4. 4.
Bodlaender HL, Downey RG, Fellows MR, Hermelin D (2009) On problems without polynomial kernels. J Comput Syst Sci 75(8):423–434
5. 5.
Crowston R, Fellows M, Gutin G, Jones M, Kim EJ, Rosamond F, Ruzsa IZ, Thomassé S, Yeo A (2014) Satisfying more than half of a system of linear equations over GF(2): a multivariate approach. J Comput Syst Sci 80(4):687–696
6. 6.
Fellows MR, Jansen BMP, Rosamond FA (2013) Towards fully multivariate algorithmics: parameter ecology and the deconstruction of computational complexity. Eur J Comb 34(3): 541–566
7. 7.
Komusiewicz C (2011) Parameterized algorithmics for network analysis: clustering & querying. PhD thesis, Technische Universität Berlin, BerlinGoogle Scholar
8. 8.
Komusiewicz C, Niedermeier R (2012) New races in parameterized algorithmics. In: MFCS ’12, Bratislava. Lecture notes in computer science, vol 7464. Springer, pp 19–30Google Scholar