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.

翻译：在查询模型中，通过子集查询预言机访问超图，我们给出了击集问题的亚线性时间算法，其参数化查询复杂度几乎紧。在参数化查询复杂度中，我们基于参数$k$（击集的大小）估计对预言机的查询次数。本文使用的子集查询预言机称为广义$d$-部独立集查询预言机（GPIS），由Bishnu等人（ISAAC'18）引入。GPIS是对Beame等人（ITCS'18和TALG'20）为估计图中边数而引入的二分独立集查询预言机（BIS）在超图上的推广。形式上，GPIS定义如下：对于$d$-一致超图$\mathcal{H}$，GPIS预言机输入$d$个两两不交的非空顶点子集$A_1, \ldots, A_d$，并回答$\mathcal{H}$中是否存在一条超边与每个集合$A_i$相交，其中$i \in \{1, 2, \ldots, d\}$。当$d=2$时，GPIS预言机即为BIS预言机。我们证明，$d$-击集问题（即$d$-一致超图的击集问题）可使用$\widetilde{\mathcal{O}}_d(k^{d} \log n)$次GPIS查询求解。此外，我们还表明，$d$-决策-击集问题（即$d$-击集问题的决策版本）可使用$\widetilde{\mathcal{O}}_d\left( \min \left\{ k^d\log n, k^{2d^2} \right\} \right)$次GPIS查询求解。我们用一个几乎匹配的参数化下界补充这些参数化上界，该下界指出，任何求解$d$-决策-击集问题的算法都需要$\Omega \left( \binom{k+d}{d} \right)$次GPIS查询。