Parameterized Query Complexity of Hitting Set using Stability of Sunflowers

In this paper, we study the query complexity of parameterized decision and optimization versions of {\sc Hitting-Set}, {\sc Vertex Cover}, {\sc Packing} , \match{} and {\sc Max-Cut}. The main focus is the query complexity of {\sc Hitting Set}. In doing so, we use an oracle known as \bis{} introduced by Beame et al.~\cite{BeameHRRS18} and its generalizations to hypergraphs. The query models considered are the \gpis{} and \gpise{} oracles : (i) the \gpis{} oracle takes as input $d$ pairwise disjoint non-empty vertex subsets $A_1, \ldots, A_d$ in a hypergraph $\cal H$ and answers whether there is a hyperedge with vertices in each $A_i$, (ii) the \gpise{} oracle takes the same input and returns a hyperedge that has vertices in each $A_i$; NULL, otherwise. The \gpis{} and \gpise{} oracles are used for the decision and optimization versions of the problems, respectively. For $d=2$, we refer \gpis{} and \gpise{} as \bis{} and \bise{}, respectively. We use color coding and queries to the oracles to generate subsamples from the hypergraph, that retain some structural properties of the original hypergraph. We use the stability of the sunflowers in a non-trivial way to do so.