Almost optimal query algorithm for hitting set using a subset query

Abstract: Given access to the hypergraph through a subset query oracle in the query model, we give sublinear time algorithms for Hitting-Set with almost tight parameterized query complexity. In parameterized query complexity, we estimate the number of queries to the oracle based on the parameter $k$, the size of the Hitting-Set. The subset query oracle we use in this paper is called Generalized $d$-partite Independent Set query oracle (GPIS) and it was introduced by Bishnu et al. (ISAAC'18). GPIS is a generalization to hypergraphs of the Bipartite Independent Set query oracle (BIS) introduced by Beame et al. (ITCS'18 and TALG'20) for estimating the number of edges in graphs. Formally, GPIS is defined as follows: GPIS oracle for a $d$-uniform hypergraph $\mathcal{H}$ takes as input $d$ pairwise disjoint non-empty subsets $A_1, \ldots, A_d$ of vertices in $\cal H$ and answers whether there is a hyperedge in $\mathcal{H}$ that intersects each set $A_i$, where $i \in {1, \, 2, \, \ldots, d}$. } For $d=2$, the GPIS oracle is nothing but BIS oracle. We show that $d$-Hitting-Set, the hitting set problem for $d$-uniform hypergraphs, can be solved using $\widetilde{\mathcal{O}}_d(k^{d} \log n)$ GPIS queries. Additionally, we also showed that $d$-Decesion-Hitting-Set, the decision version of $d$-Hitting-Set can be solved with $\widetilde{\mathcal{O}}_d\left( \min \left{ k^d\log n, k^{2d^2} \right} \right)$ {\sc GPIS} queries. We complement these parameterized upper bounds with an almost matching parameterized lower bound that states that any algorithm that solves $d$-Decesion-Hitting-Set requires $\Omega \left( \binom{k+d}{d} \right)$ GPIS queries.