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 \u00d5\\(({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|-\\text{SC}(U,F)\\). Here \\(U\\) is the set of elements, \\(F\\) is the collection of sets, and \\(\\text{SC}(U,F)\\) denotes the optimal set cover size of \\((U,F)\\). In particular, their algorithm returns a \\((1\/4,\\epsilon \\cdot |U|)\\)-multiplicative-additive estimate for this set cover problem using \\(\u00d5_{}(|F{|^(}_{7\/4)})\\) membership oracle queries (querying whether a set \\(S\\) contains an \\(e\\)), where \\(\\epsilon\\) is a fixed constant. In this work, we improve the query complexity of \\((2-\\eta )\\)-estimating the metric Steiner tree weight to \\(\u00d5_{}({n^(}_{5\/3)})\\) by showing a \\((1\/2,\\epsilon \\cdot |U|)\\)-estimate for the above set cover problem using \u00d5\\((|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.<\/p>\n","protected":false},"excerpt":{"rendered":"
We study the metric Steiner tree problem in the sublinear query model. In this problem, for a set of points in a metric space given to us by means of query access to an matrix , and a set of terminals , the goal is to find the minimum-weight subset of the edges that connects […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":[{"type":"user_nicename","value":"Sepideh Mahabadi","user_id":"40780"},{"type":"text","value":"Mohammad Roghani","user_id":0},{"type":"user_nicename","value":"Jakub Tarnawski","user_id":"38820"},{"type":"text","value":"Ali 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