Years and Authors of Summarized Original Work
2009; Bodlaender, Downey, Fellows, Hermelin
Problem Definition
Research on kernelization is motivated in two ways. First, when solving a hard (e.g., NP-hard) problem in practice, a common approach is to first preprocess the instance at hand before running more time-consuming methods (like integer linear programming, branch and bound, etc.). The following is a natural question. Suppose we use polynomial time for this preprocessing phase: what can be predicted of the size of the instance resulting from preprocessing? The theory of kernelization gives us such predictions. A second motivation comes from the fact that a decidable parameterized problem belongs to the class FPT (i.e., is fixed parameter tractable,) if and only if the problem has kernelization algorithm.
A parameterized problem is a subset of \(\varSigma ^{{\ast}}\times \mathbf{N}\), for some finite set \(\varSigma\). A kernelization algorithm (or, in short kernel) for a...
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Recommended Reading
Arnborg S, Corneil DG, Proskurowski A (1987) Complexity of finding embeddings in a k-tree. SIAM J Algebr Discret Methods 8:277–284
Binkele-Raible D, Fernau H, Fomin FV, Lokshtanov D, Saurabh S, Villanger Y (2012) Kernel(s) for problems with no kernel: on out-trees with many leaves. ACM Trans Algorithms 8(5):38
Bodlaender HL, Downey RG, Fellows MR, Hermelin D (2009) On problems without polynomial kernels. J Comput Syst Sci 75:423–434
Bodlaender HL, Jansen BMP, Kratsch S (2011) Cross-composition: a new technique for kernelization lower bounds. In: Schwentick T, Dürr C (eds) Proceedings 28th international symposium on theoretical aspects of computer science, STACS 2011, Dortmund. Schloss Dagstuhl – Leibnitz-Zentrum fuer Informatik, Leibniz International Proceedings in Informatics (LIPIcs), vol 9, pp 165–176
Bodlaender HL, Thomassé S, Yeo A (2011) Kernel bounds for disjoint cycles and disjoint paths. Theor Comput Sci 412:4570–4578
Bodlaender HL, Jansen BMP, Kratsch S (2012) Kernelization lower bounds by cross-composition. CoRR abs/1206.5941
Chen Y, Flum J, Müller M (2011) Lower bounds for kernelizations and other preprocessing procedures. Theory Comput Syst 48(4):803–839
Cygan M, Kratsch S, Pilipczuk M, Pilipczuk M, Wahlström M (2012) Clique cover and graph separation: new incompressibility results. In: Czumaj A, Mehlhorn K, Pitts AM, Wattenhofer R (eds) Proceedings of the 39th international colloquium on automata, languages and programming, ICALP 2012, Part I, Warwick. Lecture notes in computer science, vol 7391. Springer, pp 254–265
Dom M, Lokshtanov D, Saurabh S (2009) Incompressibility through colors and IDs. In: Albers S, Marchetti-Spaccamela A, Matias Y, Nikoletseas SE, Thomas W (eds) Proceedings of the 36th international colloquium on automata, languages and programming, ICALP 2009, Part I, Rhodes. Lecture notes in computer science, vol 5555. Springer, pp 378–389
Downey RG, Fellows MR (2013) Fundamentals of parameterized complexity. Texts in computer science. Springer, London
Drucker A (2012) New limits to classical and quantum instance compression. In: Proceedings of the 53rd annual symposium on foundations of computer science, FOCS 2012, New Brunswick, pp 609–618
Fortnow L, Santhanam R (2011) Infeasibility of instance compression and succinct PCPs for NP. J Comput Syst Sci 77:91–106
Gutin G, Muciaccia G, Yeo: A (2013) (Non-)existence of polynomial kernels for the test cover problem. Inf Process Lett 113:123–126
Harnik D, Naor M (2010) On the compressibility of \(\mathcal{N}\mathcal{P}\) instances and cryptographic applications. SIAM J Comput 39:1667–1713
Hermelin D, Wu X (2012) Weak compositions and their applications to polynomial lower bounds for kernelization. In: Rabani Y (ed) Proceedings of the 22nd annual ACM-SIAM symposium on discrete algorithms, SODA 2012, Kyoto. SIAM, pp 104–113
Hermelin D, Kratsch S, Soltys K, Wahlström M, Wu X (2013) A completeness theory for polynomial (turing) kernelization. In: Gutin G, Szeider S (eds) Proceedings of the 8th international symposium on parameterized and exact computation, IPEC 2013, Sophia Antipolis. Lecture notes in computer science, vol 8246. Springer, pp 202–215
Jansen BMP, Bodlaender HL (2013) Vertex cover kernelization revisited – upper and lower bounds for a refined parameter. Theory Comput Syst 53:263–299
Kratsch S (2012) Co-nondeterminism in compositions: a kernelization lower bound for a Ramsey-type problem. In: Proceedings of the 22nd annual ACM-SIAM symposium on discrete algorithms, SODA 2012, Kyoto, pp 114–122
Yap HP (1986) Some topics in graph theory. London mathematical society lecture note series, vol 108. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media New York
About this entry
Cite this entry
Bodlaender, H.L. (2016). Kernelization, Exponential Lower Bounds. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_521
Download citation
DOI: https://doi.org/10.1007/978-1-4939-2864-4_521
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-2863-7
Online ISBN: 978-1-4939-2864-4
eBook Packages: Computer ScienceReference Module Computer Science and Engineering