Towards optimal and expressive kernelization for d-Hitting Set

Abstract. A sunflower in a hypergraph is a set of hyperedges pairwise in-tersecting 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(kd) hy-peredges 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(kd−1) with addi-tional processing in O(k1.5d) time—nontrivially combining the sunflower technique with known problem kernels due to Abu-Khzam and Moser.