We study the metric Steiner tree problem in the sublinear query model. In this problem, for a set of $n$ points $V$ in a metric space given to us by means of query access to an $n\times n$ matrix $w$, and a set of terminals $T\subseteq V$, the goal is to find the minimum-weight subset of the edges that connects all the terminal vertices. Recently, Chen, Khanna and Tan [SODA'23] gave an algorithm that uses $\widetilde{O}(n^{13/7})$ queries and outputs a $(2-\eta)$-estimate of the metric Steiner tree weight, where $\eta>0$ is a universal constant. A key component in their algorithm is a sublinear algorithm for a particular set cover problem where, given a set system $(U, F)$, the goal is to provide a multiplicative-additive estimate for $|U|-\textsf{SC}(U, F)$. Here $U$ is the set of elements, $F$ is the collection of sets, and $\textsf{SC}(U, F)$ denotes the optimal set cover size of $(U, F)$. In particular, their algorithm returns a $(1/4, \varepsilon\cdot|U|)$-multiplicative-additive estimate for this set cover problem using $\widetilde{O}(|F|^{7/4})$ membership oracle queries (querying whether a set $S$ contains an $e$), where $\varepsilon$ is a fixed constant. In this work, we improve the query complexity of $(2-\eta)$-estimating the metric Steiner tree weight to $\widetilde{O}(n^{5/3})$ by showing a $(1/2, \varepsilon \cdot |U|)$-estimate for the above set cover problem using $\widetilde{O}(|F|^{5/3})$ membership queries. To design our set cover algorithm, we estimate the size of a random greedy maximal matching for an auxiliary multigraph that the algorithm constructs implicitly, without access to its adjacency list or matrix.

翻译：我们在亚线性查询模型中研究度量Steiner树问题。该问题中，给定通过查询访问$n\times n$矩阵$w$获得的度量空间中$n$个点集$V$，以及终端点集$T\subseteq V$，目标是找到连接所有终端顶点的最小权重边子集。最近，Chen、Khanna和Tan [SODA'23]提出了一种使用$\widetilde{O}(n^{13/7})$次查询的算法，可输出度量Steiner树权重的$(2-\eta)$近似估计，其中$\eta>0$为通用常数。其算法的关键组成部分是针对特定集合覆盖问题的亚线性算法：给定集合系统$(U, F)$，目标是为$|U|-\textsf{SC}(U, F)$提供乘加性估计。此处$U$为元素集，$F$为集合族，$\textsf{SC}(U, F)$表示$(U, F)$的最优集合覆盖大小。特别地，他们的算法通过$\widetilde{O}(|F|^{7/4})$次成员资格查询（查询集合$S$是否包含元素$e$），返回该集合覆盖问题的$(1/4, \varepsilon\cdot|U|)$乘加性估计，其中$\varepsilon$为固定常数。本工作中，我们通过使用$\widetilde{O}(|F|^{5/3})$次成员资格查询为上述集合覆盖问题提供$(1/2, \varepsilon \cdot |U|)$估计，将度量Steiner树权重的$(2-\eta)$近似估计的查询复杂度改进至$\widetilde{O}(n^{5/3})$。为设计集合覆盖算法，我们通过隐式构建辅助多重图（无需访问其邻接表或矩阵），估计该图中随机贪婪最大匹配的规模。