I  C  Id

In parameterized complexity each problem instance comes with a parameter k and the parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance down to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the non-existence of polynomial kernels for specific problems has been developed by Bodlaender et al. [6] and Fortnow and Santhanam [15]. With few exceptions, all known kernelization lower bounds result have been obtained by directly applying this framework. In this paper we show how to combine these results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems. Below we give a summary of our main results. All our results are under the assumption that the polynomial hierarchy does not collapse to the third level. • We show that the S T problem parameterized by the number of terminals and solution size, and the C V C and C V C problems do not admit a polynomial kernel. The two latter results are surprising because the closely related V C problem admits a kernel of size 2k. • Alon and Gutner obtain a kpoly(h) kernel for D S  H-M F G parameterized by h = |H| and k, and ask whether kernels of smaller size exist [3]. We partially resolve this question by showing that D S  H-M F G does not admit a kernel with size polynomial in k + h. • Harnik and Naor obtain a “compression algorithm” for the S S S problem [19]. We show that their algorithm is essentially optimal by showing that the instances can not be compressed further. • The H S and S C problems are among the most studied problems in algorithms. Both problems admit a kernel of size kO(d) when parameterized by solution size k and maximum set size d. We show that neither of them, along with the U C and B R D S problems, admit a polynomial kernel. The existence of polynomial kernels for several of the problems mentioned above were open problems explicitly stated in the literature [3, 4, 17, 18, 22]. Many of our results also rule out the existence of compression algorithms, a notion similar to kernelization defined by Harnik and Naor [19], for the problems in question. Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany. [email protected] Department of Informatics, University of Bergen, 5020 Bergen, Norway. {daniello|Saket.Saurabh}@ii.uib.no