Lower Bounds for Kernelizations and Other Preprocessing Procedures

Abstract

We first present a method to rule out the existence of strong polynomial kernelizations of parameterized problems under the hypothesis P $\ne$ 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 [1,6] by refining the central lemma of their proof method, a lemma due to Fortnow and Santhanam. In particular, assuming that PH $\ne \Sigma^{\rm {P}}_3$, 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)·|x|^1 − ε (here ε is any given real number greater than zero).