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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
Tree Deletion Set has a Polynomial Kernel (but no OPT O(1) approximation
 CoRR
, 2013
"... In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G \ S is a tree. The problem is NPcomplete and even NPhard to approximate within any factor of OPTc for any constant c. In ..."
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In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G \ S is a tree. The problem is NPcomplete and even NPhard to approximate within any factor of OPTc for any constant c. In this paper we give an O(k5) size kernel for the Tree Deletion Set problem. An appealing feature of our kernelization algorithm is a new reduction rule, based on system of linear equations, that we use to handle the instances on which Tree Deletion Set is hard to approximate.
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.