Parameterized Query Complexity of Hitting Set using Stability of Sunflowers
share
In this paper, we study the query complexity of parameterized decision and optimization versions of Hitting-Set, Vertex Cover, Packing , and Max-Cut. The main focus is the query complexity of Hitting Set. In doing so, we use an oracle known as introduced by Beame et al. BeameHRRS18 and its generalizations to hypergraphs. The query models considered are the and oracles : (i) the oracle takes as input d pairwise disjoint non-empty vertex subsets A_1, ..., A_d in a hypergraph H and answers whether there is a hyperedge with vertices in each A_i, (ii) the oracle takes the same input and returns a hyperedge that has vertices in each A_i; NULL, otherwise. The and oracles are used for the decision and optimization versions of the problems, respectively. For d=2, we refer and as and , 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.
READ FULL TEXT