Sublinear Metric Steiner Tree via Improved Bounds for Set Cover

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.