Lower Bounds for Kernelizations and Other Preprocessing Procedures

Abstract

We first present a method to rule out the existence of parameter non-increasing polynomial kernelizations of parameterized problems under the hypothesis P≠NP. This method is applicable, for example, to the problem Sat parameterized by the number of variables of the input formula. Then we obtain further improvements of corresponding results in (Bodlaender et al. in Lecture Notes in Computer Science, vol. 5125, pp. 563–574, Springer, Berlin, 2008; Fortnow and Santhanam in Proceedings of the 40th ACM Symposium on the Theory of Computing (STOC’08), ACM, New York, pp. 133–142, 2008) by refining the central lemma of their proof method, a lemma due to Fortnow and Santhanam. In particular, assuming 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 self-reductions that assign to every instance x with parameter k an instance y with |y|=k O(1)⋅|x|1−e (here e is any given real number greater than zero). We give various applications of these results. On the structural side we prove several results clarifying the relationship between the different notions of preprocessing procedures, namely the various notions of kernelizations, self-reductions and compressions.