Years and Authors of Summarized Original Work
2010; Dell, van Melkebeek
Problem Definition
The work of Dell and van Melkebeek [4] refines the framework for lower bounds for kernelization introduced by Bodlaender et al. [1] and Fortnow and Santhanam [6]. The main contribution is that their results yield a framework for proving polynomial lower bounds for kernelization rather than ruling out all polynomial kernels for a problem; this, for the first time, gives a technique for proving that some polynomial kernelizations are actually best possible, modulo reasonable complexity assumptions. A further important aspect is that, rather than studying kernelization directly, the authors give lower bounds for a far more general oracle communication protocol. In this way, they also obtain strong lower bounds for sparsification, lossy compression (in the sense of Harnik and Naor [7]), and probabilistically checkable proofs (PCPs).
oracle communication protocols, let us first recall the following. A
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Recommended Reading
Bodlaender HL, Downey RG, Fellows MR, Hermelin D (2009) On problems without polynomial kernels. J Comput Syst Sci 75(8):423–434
Cygan M, Grandoni F, Hermelin D (2013) Tight kernel bounds for problems on graphs with small degeneracy – (extended abstract). In: Bodlaender HL, Italiano GF (eds) ESA. Lecture notes in computer science, vol 8125. Springer, pp 361–372
Dell H, Marx D (2012) Kernelization of packing problems. In: SODA, Kyoto. SIAM, pp 68–81
Dell H, van Melkebeek D (2010) Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Schulman LJ (ed) STOC, Cambridge. ACM, pp 251–260
Dell H, van Melkebeek D (2010) Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. Electron Colloq Comput Complex 17:38
Fortnow L, Santhanam R (2011) Infeasibility of instance compression and succinct PCPs for NP. J Comput Syst Sci 77(1):91–106
Harnik D, Naor M (2010) On the compressibility of \(\mathcal{N}\mathcal{P}\) instances and cryptographic applications. SIAM J Comput 39(5):1667–1713
Hermelin D, Wu X (2012) Weak compositions and their applications to polynomial lower bounds for kernelization. In: SODA, Kyoto. SIAM, pp 104–113
Kratsch S (2012) Co-nondeterminism in compositions: a kernelization lower bound for a Ramsey-type problem. In: SODA, Kyoto. SIAM, pp 114–122
Kratsch S, Pilipczuk M, Rai A, Raman V (2012) Kernel lower bounds using co-nondeterminism: finding induced hereditary subgraphs. In: Fomin FV, Kaski P (eds) SWAT, Helsinki. Lecture notes in computer science, vol 7357. Springer, pp 364–375
Kratsch S, Philip G, Ray S (2014) Point line cover: the easy kernel is essentially tight. In: SODA, Portland. SIAM, pp 1596–1606
Thomassé S (2010) A quadratic kernel for feedback vertex set. ACM Trans Algorithms 6(2), 32:1–32:8
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Kratsch, S. (2016). Kernelization, Polynomial Lower Bounds. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_522
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DOI: https://doi.org/10.1007/978-1-4939-2864-4_522
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