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

Towards Optimal and Expressive Kernelization for d-Hitting Set

  • René van Bevern


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.


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



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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