Kernelization is a formalization of efficient preprocessing, aimed mainly at combinatorially hard problems. Empirically, preprocessing is highly suc-cessful in practice, e.g., in state-of-the-art SAT and ILP solvers. The notion of kernelization from parameterized complexity makes it possible to rigor-ously prove upper and lower bounds on, e.g., the maximum output size of a preprocessing in terms of one or more problem-specific parameters. This avoids the often-raised issue that we should not expect an efficient algorithm that provably shrinks every instance of any NP-hard problem. In this survey, we give a general introduction to the area of kernelization and then discuss some recent developments. After the introductory material we attempt a reasonably self-contained update and introduction on the fol-lowing topics: (1) Lower bounds for kernelization, taking into account the recent progress on the and-conjecture. (2) The use of matroids and repre-sentative sets for kernelization. (3) Turing kernelization, i.e., understanding preprocessing that adaptively or non-adaptively creates a large number of small outputs. 1

Mathematical logic and computational complexity have close connections that can be traced to the roots of computability theory and the classical decision problem. In the context of complexity, some well-known fundamental problems are: satisfiability testing of formulas (in some logic), proof complexity, and the complexity of checking if a given model satisfies a given formula. The Model Checking problem, which is the topic of the present article, is also of practical relevance since efficient model checking algorithms for temporal/modal logics are useful in formal verification. In their excellent and detailed survey, Arne Meier, Martin Mundhenk, Julian-Steffen Müller, and Heribert Vollmer tell us about the complexity of model checking for various logics: temporal, modal and hybrid and their many frag-ments. Their article brings out the intricate structures involved in the reduc-tions and the effectiveness of standard complexity classes in capturing the complexity of model checking.

We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, |V (H) | is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on node-disjoint paths, and computing minors seems easier in sub-cubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph G of treewidth k, computes a topological minor H of G such that (i) the treewidth of H is Ω(k/polylog(k)); (ii) |V (H) | = O(k4); and (iii) the maximum vertex degree in H is 3. The running time of the algorithm is polynomial in |V (G) | and k. Our result is in contrast to the known fact that unless NP ⊆ coNP/poly, treewidth does not admit polynomial-size kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves node-disjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small good-quality vertex-cut sparsifiers that are also minors of the original graph.