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Towards Optimal and Expressive Kernelization for d-Hitting Set

René van Bevern (TU Berlin)

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 (of cardinality at most 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 theory, we show a problem kernel with asymptotically optimal size (unless coNP in NP/poly). 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 a known problem kernel that uses crown reductions.

12.01.2012 16:15
René van Bevern
FR 6510

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