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Algorithmica

, Volume 70, Issue 1, pp 129–147 | Cite as

Towards Optimal and Expressive Kernelization for d-Hitting Set

  • René van Bevern
Article

Abstract

A sunflower in a hypergraph is a set of hyperedges pairwise intersecting in exactly the same vertex set. Sunflowers are a useful tool in polynomial-time data reduction for problems formalizable as d-Hitting Set, the problem of covering all hyperedges (whose cardinality is bounded from above by a constant d) of a hypergraph by at most k vertices. Additionally, in fault diagnosis, sunflowers yield concise explanations for “highly defective structures”.

We provide a linear-time algorithm that, by finding sunflowers, transforms an instance of d-Hitting Set into an equivalent instance comprising at most O(k d ) hyperedges and vertices. In terms of parameterized complexity, we show a problem kernel with asymptotically optimal size (unless \(\operatorname {coNP}\subseteq \operatorname {NP/poly}\)) and provide experimental results that show the practical applicability of our algorithm.

Finally, we show that the number of vertices can be reduced to O(k d−1) with additional processing in O(k 1.5d ) time—nontrivially combining the sunflower technique with problem kernels due to Abu-Khzam and Moser.

Keywords

Parameterized algorithmics Linear-time data reduction Vertex cover in hypergraphs Fault diagnosis Sunflower lemma Algorithm engineering 

Notes

Acknowledgements

The author is very thankful to Rolf Niedermeier and the anonymous referees for many valuable comments.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany

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