A Basic Parameterized Complexity Primer

Downey, Rodney G.

doi:10.1007/978-3-642-30891-8_9

Cited by 6 publications

(4 citation statements)

References 76 publications

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“…However, for instance checking the order of the polynomial is constant (Observation 17), and for conjunctive query entailment the order of the polynomial grows with the size of the query q (Corollary 19). This is a difference with enormous effects on the practicality of algorithms solving these problems (see, e.g., Downey 2012).…”

Section: Interpretation Of the Resultsmentioning

confidence: 99%

“…Suppose, for example, that a problem is solvable in time n k in the worst case. Then already for n = 100 and k = 10, it could take ages to solve this problem (see, e.g., Downey 2012).…”

Section: Parameterized Complexity Theorymentioning

confidence: 99%

“…The parameter k can be used to measure various types of structure that are present in the problem input. Parameterized complexity theory has grown into a large and thriving research community over the last few decades (see, e.g., Bodlaender et al 2012; Downey 2012;Downey and Fellows 2013). Most results and techniques in parameterized complexity theory revolve around the notion of fixed-parameter tractability-a relaxation of polynomialtime solvability based on running times of the form f (k) • n O(1) , for some computable function f (possibly exponential or worse).…”

Section: Introductionmentioning

confidence: 99%

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A Parameterized Complexity View on Description Logic Reasoning

Haan¹

2018

Preprint

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Description logics are knowledge representation languages that have been designed to strike a balance between expressivity and computational tractability. Many different description logics have been developed, and numerous computational problems for these logics have been studied for their computational complexity. However, essentially all complexity analyses of reasoning problems for description logics use the one-dimensional framework of classical complexity theory. The multi-dimensional framework of parameterized complexity theory is able to provide a much more detailed image of the complexity of reasoning problems. In this paper we argue that the framework of parameterized complexity has a lot to offer for the complexity analysis of description logic reasoning problems-when one takes a progressive and forward-looking view on parameterized complexity tools. We substantiate our argument by means of three case studies. The first case study is about the problem of concept satisfiability for the logic ALC with respect to nearly acyclic TBoxes. The second case study concerns concept satisfiability for ALC concepts parameterized by the number of occurrences of union operators and the number of occurrences of full existential quantification. The third case study offers a critical look at data complexity results from a parameterized complexity point of view. These three case studies are representative for the wide range of uses for parameterized complexity methods for description logic problems.

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Section: Interpretation Of the Resultsmentioning

confidence: 99%

“…Suppose, for example, that a problem is solvable in time n k in the worst case. Then already for n = 100 and k = 10, it could take ages to solve this problem (see, e.g., Downey 2012).…”

Section: Parameterized Complexity Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Parameterized Complexity View on Description Logic Reasoning

Haan¹

2018

Preprint

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“…Recall that a parameterized problem is fixed-parameter tractable (FPT) with a given parameterization if there is an algorithm with running time f (k) · n O (1) , where n is the size of the instance, k is the value of the parameter associated with the instance, and f is an arbitrary computable function depending only on the parameter k (see the monographs [44,50,87] or the survey [41] in this volume for more background). That is, the problem can be solved in polynomial time for every fixed value of the parameter k and the exponent does not depend on the parameter.…”

Section: The Optimality Programmentioning

confidence: 99%

What’s Next? Future Directions in Parameterized Complexity

Marx

2012

The Multivariate Algorithmic Revolution and Beyond

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Abstract. The progress in parameterized complexity has been very significant in recent years, with new research questions and directions, such as kernelization lower bounds, appearing and receiving considerable attention. This speculative article tries to identify new directions that might become similar hot topics in the future. First, we point out that the search for optimality in parameterized complexity already has good foundations, but lots of interesting work can be still done in this area. The systematic study of kernelization became a very successful research direction in recent years. We look at what general conclusions one can draw from these results and we argue that the systematic study of other algorithmic techniques should be modeled after the study of kernelization. In particular, we set up a framework for understanding which problems can be solved by branching algorithms. Finally, we discuss that the domain of directed graph problems is a challenging area which can potentially see significant progress in the following years.

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