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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
Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask)
, 2014
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A linear kernel for planar total dominating set. Manuscript available at arxiv.org/abs/1211.0978
, 2012
"... Abstract. A total dominating set of a graph G = (V,E) is a subset D ⊆ V such that every vertex in V is adjacent to some vertex in D. Finding a total dominating set of minimum size is NPhard on planar graphs and W [2]complete on general graphs when parameterized by the solution size. By the metath ..."
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Abstract. A total dominating set of a graph G = (V,E) is a subset D ⊆ V such that every vertex in V is adjacent to some vertex in D. Finding a total dominating set of minimum size is NPhard on planar graphs and W [2]complete on general graphs when parameterized by the solution size. By the metatheorem of Bodlaender et al. [FOCS 2009], it follows that there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM 2004], we provide an explicit linear kernel for Total Dominating Set on planar graphs. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominat
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.
Kernelization and Sparseness: the case of Dominating Set∗
, 2014
"... The search for linear kernels for the Dominating Set problem on classes of graphs of a topological nature has been one of the leading trends in kernelization in recent years. Following the fundamental work of Alber et al. [2] that established a linear kernel for the problem on planar graphs, linear ..."
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The search for linear kernels for the Dominating Set problem on classes of graphs of a topological nature has been one of the leading trends in kernelization in recent years. Following the fundamental work of Alber et al. [2] that established a linear kernel for the problem on planar graphs, linear kernels have been given for boundedgenus graphs [4], apexminorfree graphs [15], Hminorfree graphs [16], and Htopologicalminorfree graphs [17]. These generalizations are based on bidimensionality and powerful decomposition theorems for Hminorfree graphs and Htopologicalminorfree graphs of Robertson and Seymour [28] and of Grohe and Marx [22]. In this work we investigate a new approach to kernelization algorithms for Dominating Set on sparse graph classes. The approach is based on the theory of bounded expansion and nowhere dense graph classes, developed in the recent years by Nešetřil and Ossona de Mendez, among others. More precisely, we prove that Dominating Set admits a linear kernel on any hereditary graph class of bounded expansion and an almost linear kernel on any hereditary nowhere dense graph class. Since the class of Htopologicalminorfree graphs has bounded expansion, our results strongly generalize all the above mentioned works on kernelization of Dominating Set.
(Meta) Kernelization
, 2013
"... In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k, while preserving the answer. In this work we give two metatheorems on kernelzation. ..."
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In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k, while preserving the answer. In this work we give two metatheorems on kernelzation. The first theorem says that all problems expressible in Counting Monadic Second Order Logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus. Our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus. These theorems unify and extend all previously known kernelization results for planar graph problems.