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 ⊆ 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 known problem kernels due to Abu-Khzam and Moser.
A linear-time algorithm REF produces a kernel with O(k d ) hyperedges; with additional O(k 1.5d ) time the number of vertices can be reduced to
Towards Optimal and Expressive Kernelization for d-Hitting Set
Algorithmica 70(1):129-147, 2014
Mathematics Computer Science