Towards optimal and expressive kernelization for d-hitting set

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22 Citations (Scopus)

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 coNP ⊂/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.

Original languageEnglish
Pages (from-to)129-147
Number of pages19
JournalAlgorithmica
Volume70
Issue number1
DOIs
Publication statusPublished - 1 Jan 2014
Externally publishedYes

Keywords

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

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