Abstract
Kernelization—a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems—plays a central role in parameterized complexity and has triggered an extensive line of research. In this paper we consider a restricted yet natural variant of kernelization, namely strict kernelization, where one is not allowed to increase the parameter of the reduced instance (the kernel) by more than an additive constant. Building on earlier work of Chen, Flum, and Müller [CiE 2009, Theory Comput. Syst. 2011], we underline the applicability of their framework by showing that a variety of fixed-parameter tractable problems, including graph problems and Turing machine computation problems, does not admit strict polynomial kernels under the weaker assumption of \( P {}\ne NP {}\). Finally, we study a relaxation of the notion of strict kernels.
Work initiated by the research retreat of the Theoretical Computer Science group of the Universität of Tübingen in Sulz (Neckar), September 2016.
T. Fluschnik—Supported by the DFG, project DAMM (NI 369/13) and project TORE (NI 369/18).
D. Hermelin—Supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement number 631163.11, and by the ISRAEL SCIENCE FOUNDATION (grant No. 551145/14). Also supported by a DFG Mercator fellowship, project DAMM (NI 369/13) while staying at TU Berlin (August 2016).
A. Krebs—Supported by the DFG Emmy Noether program (KR 4042/2).
H. Molter—Partially supported by the DFG, project DAPA (NI 369/12).
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Notes
- 1.
For a complete list of problem definitions we refer to a long version [12] of the paper.
- 2.
Full proofs of results marked with (\(\star \)) are deferred to a long version [12] of the paper.
- 3.
Note that Chen et al. [8, Proposition 3.3] presented an artificial parameterized problem admitting a polynomial kernel but no strict polynomial kernel.
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Fernau, H., Fluschnik, T., Hermelin, D., Krebs, A., Molter, H., Niedermeier, R. (2018). Diminishable Parameterized Problems and Strict Polynomial Kernelization. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_17
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