{"id":1133744,"date":"2025-03-05T18:41:53","date_gmt":"2025-03-06T02:41:53","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=1133744"},"modified":"2025-03-05T18:41:55","modified_gmt":"2025-03-06T02:41:55","slug":"sublinear-metric-steiner-tree-via-improved-bounds-for-set-cover","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/sublinear-metric-steiner-tree-via-improved-bounds-for-set-cover\/","title":{"rendered":"Sublinear Metric Steiner Tree via Improved Bounds for Set Cover"},"content":{"rendered":"

We study the metric Steiner tree problem in the sublinear query model. In this problem, for a set of n<\/span><\/span><\/span><\/span> points V<\/span><\/span><\/span><\/span> in a metric space given to us by means of query access to an n<\/span>\u00d7<\/span>n<\/span><\/span><\/span><\/span> matrix w<\/span><\/span><\/span><\/span>, and a set of terminals T<\/span>\u2286<\/span>V<\/span><\/span><\/span><\/span>, the goal is to find the minimum-weight subset of the edges that connects all the terminal vertices.
\nRecently, Chen, Khanna and Tan [SODA’23] gave an algorithm that uses \u00d5(<\/span>n^(<\/span>13<\/span>\/<\/span><\/span><\/span>7)<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span> queries and outputs a (<\/span>2<\/span>\u2212<\/span>\u03b7<\/span>)<\/span><\/span><\/span><\/span>-estimate of the metric Steiner tree weight, where \u03b7<\/span>><\/span>0<\/span><\/span><\/span><\/span> 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 (<\/span>U<\/span>,<\/span>F<\/span>)<\/span><\/span><\/span><\/span>, the goal is to provide a multiplicative-additive estimate for |<\/span><\/span><\/span>U<\/span>|<\/span><\/span><\/span>\u2212<\/span>SC<\/span><\/span><\/span>(<\/span>U<\/span>,<\/span>F<\/span>)<\/span><\/span><\/span><\/span>. Here U<\/span><\/span><\/span><\/span> is the set of elements, F<\/span><\/span><\/span><\/span> is the collection of sets, and SC<\/span><\/span><\/span>(<\/span>U<\/span>,<\/span>F<\/span>)<\/span><\/span><\/span><\/span> denotes the optimal set cover size of (<\/span>U<\/span>,<\/span>F<\/span>)<\/span><\/span><\/span><\/span>. In particular, their algorithm returns a (<\/span>1<\/span>\/<\/span><\/span><\/span>4<\/span>,<\/span>\u03b5<\/span>\u22c5<\/span>|<\/span><\/span><\/span>U<\/span>|<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>-multiplicative-additive estimate for this set cover problem using \u00d5<\/span><\/span><\/span><\/span>(<\/span>|<\/span><\/span><\/span>F<\/span>|^(<\/span><\/span><\/span>7<\/span>\/<\/span><\/span><\/span>4)<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span> membership oracle queries (querying whether a set S<\/span><\/span><\/span><\/span> contains an e<\/span><\/span><\/span><\/span>), where \u03b5<\/span><\/span><\/span><\/span> is a fixed constant.
\nIn this work, we improve the query complexity of (<\/span>2<\/span>\u2212<\/span>\u03b7<\/span>)<\/span><\/span><\/span><\/span>-estimating the metric Steiner tree weight to \u00d5<\/span><\/span><\/span><\/span>(<\/span>n^(<\/span>5<\/span>\/<\/span><\/span><\/span>3)<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span> by showing a (<\/span>1<\/span>\/<\/span><\/span><\/span>2<\/span>,<\/span>\u03b5<\/span>\u22c5<\/span>|<\/span><\/span><\/span>U<\/span>|<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>-estimate for the above set cover problem using \u00d5(<\/span>|<\/span><\/span><\/span>F<\/span>|^(<\/span><\/span><\/span>5<\/span>\/<\/span><\/span><\/span>3)<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span> 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 n points V in a metric space given to us by means of query access to an n\u00d7n matrix w, and a set of terminals T\u2286V, the goal is to find the minimum-weight subset of the 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