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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 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 rigo ..."
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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
Tight Approximation Bounds for Vertex Cover on Dense k-Partite Hypergraphs
"... We establish almost tight upper and lower approximation bounds for the Vertex Cover problem on dense k-partite hypergraphs. ..."
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We establish almost tight upper and lower approximation bounds for the Vertex Cover problem on dense k-partite hypergraphs.
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 well-known 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 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.
Multi-Tuple Deletion Propagation: Approximations and Complexity
, 2013
"... This paper studies the computational complexity of the classic problem of deletion propagation in a relational database, where tuples are deleted from the base relations in order to realize a desired deletion of tuples from the view. Such an operation may result in a (sometimes unavoidable) side eff ..."
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This paper studies the computational complexity of the classic problem of deletion propagation in a relational database, where tuples are deleted from the base relations in order to realize a desired deletion of tuples from the view. Such an operation may result in a (sometimes unavoidable) side effect: deletion of additional tuples from the view, besides the intentionally deleted ones. The goal is to minimize the side effect. The complexity of this problem has been well studied in the case where only a single tuple is deleted from the view. However, only little is known within the more realistic scenario of multi-tuple deletion, which is the topic of this paper. The class of conjunctive queries (CQs) is among the most well studied in the literature, and we focus here on views defined by CQs that are self-join free (sjf-CQs). Our main result is a trichotomy in complexity, classifying all sjf-CQs into three categories: those for which the problem is in polynomial time, those for which the problem is NP-hard but polynomial-time approximable (by a constantfactor), and those for which even an approximation (by any factor) is NP-hard to obtain. A corollary of this trichotomy is a dichotomy in the complexity of deciding whether a sideeffect-free solution exists, in the multi-tuple case. We further extend the full classification to accommodate the presence of a constant upper bound on the number of view tuples to delete, and the presence of functional dependencies. Finally, we establish (positive and negative) complexity results on approximability for the dual problem of maximizing the number of view tuples surviving (rather than minimizing the side effect incurred in) the deletion propagation.
