Uniform Kernelization Complexity of Hitting Forbidden Minors

The F-Minor-Free Deletion problem asks, for a fixed set F and an input consisting of a graph G and integer k, whether k vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. This paper analyzes to what extent provably effective and efficient preprocessing is possible for F-Minor-Free Deletion. Fomin et al. (FOCS 2012) showed that the special case Planar F-Deletion (when F contains at least one planar graph) has a kernel of size f(F) * k^{g(F)} for some functions f and g. The degree g of the polynomial grows very quickly; it is not even known to be computable. Fomin et al. left open whether Planar F-Deletion has kernels whose size is uniformly polynomial, i.e., of the form f(F) * k^c for some universal constant c that does not depend on F. Our results in this paper are twofold. (1) We prove that some Planar F-Deletion problems do not have uniformly polynomial kernels (unless NP is in coNP/poly). In particular, we prove that Treewidth-Eta Deletion does not have a kernel with O(k^{eta/4} - eps) vertices for any eps > 0, unless NP is in coNP/poly. In fact, we even prove the kernelization lower bound for the larger parameter vertex cover number. This resolves an open problem of Cygan et al. (IPEC 2011). It is a natural question whether further restrictions on F lead to uniformly polynomial kernels. However, we prove that even when F contains a path, the degree of the polynomial must, in general, depend on the set F. (2) A canonical F-Minor-Free Deletion problem when F contains a path is Treedepth-eta Deletion: can k vertices be removed to obtain a graph of treedepth at most eta? We prove that Treedepth-eta Deletion admits uniformly polynomial kernels with O(k^6) vertices for every fixed eta. In order to develop the kernelization we prove several new results about the structure of optimal treedepth-decompositions.