Lower Bounds for Kernelizations

We first present a method to rule out the existence of strong p oly- nomial kernelizations of parameterized problems under the hypothesis P 6 NP. For example, this method is applicable to the problem SAT parameterized by the number of variables of the input formula. Then we obtain further improvements of corresponding results in (5, 7) by refining the central lem ma 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 clas- sical problem does not have polynomial reductions to itself that assign to every instance x with parameter k an instance y with |y| kO � | x| " (here " is any given real number greater than zero).