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Lower bounds for kernelizations and other preprocessing procedures. (English) Zbl 1268.68084
Ambos-Spies, Klaus (ed.) et al., Mathematical theory and computational practice. 5th conference on computability in Europe, CiE 2009, Heidelberg, Germany, July 19–24, 2009. Proceedings. Berlin: Springer (ISBN 978-3-642-03072-7/pbk). Lecture Notes in Computer Science 5635, 118-128 (2009).
Summary: We first present a method to rule out the existence of strong polynomial kernelizations of parameterized problems under the hypothesis \(\text{P} \neq \text{NP}\). This method is applicable, for example, to the problem SAT parameterized by the number of variables of the input formula. Then we obtain improvements of related results in [H. L. Bodlaender et al., Lect. Notes Comput. Sci. 5125, 563–574 (2008; Zbl 1153.68554); L. Fortnow and R. Santhanam, in: Proceedings of the 40th annual ACM symposium on theory of computing, Victoria, Canada, STOC’08. New York, NY: Association for Computing Machinery. 133–142 (2008; Zbl 1231.68133)] by refining the central lemma of their proof method, a lemma due to Fortnow and Santhanam. In particular, assuming that \(\text{PH} \neq \Sigma^{\text{P}}_3\), i.e., that the polynomial hierarchy does not collapse to its third level, we show that every parameterized problem with a “linear OR” and with NP-hard underlying classical problem does not have polynomial reductions to itself that assign to every instance \(x\) with parameter \(k\) an instance \(y\) with \(|y| = k^{O(1)} \cdot |x|^{1 - \epsilon}\) (here \(\epsilon \) is any given real number greater than zero).
For the entire collection see [Zbl 1192.68004].

MSC:
68Q25 Analysis of algorithms and problem complexity
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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