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 
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Recommended Reading

  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):774–789MathSciNetCrossRefMATHGoogle Scholar
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    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):721–748CrossRefGoogle Scholar
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    Bodlaender HL, Downey RG, Fellows MR, Hermelin D (2009) On problems without polynomial kernels. J Comput Syst Sci 75(8):423–434MathSciNetCrossRefMATHGoogle Scholar
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    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):687–696CrossRefMATHGoogle Scholar
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    Fellows MR, Jansen BMP, Rosamond FA (2013) Towards fully multivariate algorithmics: parameter ecology and the deconstruction of computational complexity. Eur J Comb 34(3): 541–566MathSciNetCrossRefMATHGoogle Scholar
  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

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institut für Softwaretechnik und Theoretische InformatikBerlinGermany