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Recent developments in kernelization: A survey
"... Kernelization is a formalization of efficient preprocessing, aimed mainly at combinatorially hard problems. Empirically, preprocessing is highly successful in practice, e.g., in stateoftheart SAT and ILP solvers. The notion of kernelization from parameterized complexity makes it possible to rigo ..."
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Kernelization is a formalization of efficient preprocessing, aimed mainly at combinatorially hard problems. Empirically, preprocessing is highly successful in practice, e.g., in stateoftheart SAT and ILP solvers. The notion of kernelization from parameterized complexity makes it possible to rigorously prove upper and lower bounds on, e.g., the maximum output size of a preprocessing in terms of one or more problemspecific parameters. This avoids the oftenraised issue that we should not expect an efficient algorithm that provably shrinks every instance of any NPhard 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 selfcontained update and introduction on the following topics: (1) Lower bounds for kernelization, taking into account the recent progress on the andconjecture. (2) The use of matroids and representative sets for kernelization. (3) Turing kernelization, i.e., understanding preprocessing that adaptively or nonadaptively creates a large number of small outputs. 1
The Computational Complexity Column
"... 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 wellknown fundamental problems are: satisfiability testing of formulas (in some logic), proof complex ..."
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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 wellknown 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, JulianSteffen Müller, and Heribert Vollmer tell us about the complexity of model checking for various logics: temporal, modal and hybrid and their many fragments. Their article brings out the intricate structures involved in the reductions and the effectiveness of standard complexity classes in capturing the complexity of model checking.
Degree3 Treewidth Sparsifiers
, 2014
"... 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 nodedisj ..."
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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 nodedisjoint paths, and computing minors seems easier in subcubic 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 polynomialsize kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves nodedisjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small goodquality vertexcut sparsifiers that are also minors of the original graph.