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Sublinear Metric Steiner Tree via Improved Bounds for Set Cover

Authors Sepideh Mahabadi , Mohammad Roghani , Jakub Tarnawski , Ali Vakilian

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Author Details

Sepideh Mahabadi
  • Microsoft Research, Redmond, WA, USA
Mohammad Roghani
  • Stanford University, CA, USA
Jakub Tarnawski
  • Microsoft Research, Redmond, WA, USA
Ali Vakilian
  • Toyota Technological Institute at Chicago (TTIC), Il, USA

Acknowledgements

This work was done while Mohammad Roghani was an intern at Microsoft Research. The work was conducted in part while Sepideh Mahabadi and Ali Vakilian were long-term visitors at the Simons Institute for the Theory of Computing as part of the Sublinear Algorithms program.

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Sepideh Mahabadi, Mohammad Roghani, Jakub Tarnawski, and Ali Vakilian. Sublinear Metric Steiner Tree via Improved Bounds for Set Cover. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 74:1-74:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.74

Abstract

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× n matrix w, and a set of terminals T ⊆ 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 Õ(n^{13/7}) queries and outputs a (2-η)-estimate of the metric Steiner tree weight, where η > 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|-SC(𝒰, ℱ). Here 𝒰 is the set of elements, ℱ is the collection of sets, and SC(𝒰, ℱ) denotes the optimal set cover size of (𝒰, ℱ). In particular, their algorithm returns a (1/4, ε⋅|𝒰|)-multiplicative-additive estimate for this set cover problem using Õ(|ℱ|^{7/4}) membership oracle queries (querying whether a set S ∈ 𝒮 contains an element e ∈ 𝒰), where ε is a fixed constant. 
In this work, we improve the query complexity of (2-η)-estimating the metric Steiner tree weight to Õ(n^{5/3}) by showing a (1/2, ε⋅|𝒰|)-estimate for the above set cover problem using Õ(|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. Previous analyses of random greedy maximal matching have focused on simple graphs, assuming access to their adjacency list or matrix. To address this, we extend the analysis of Behnezhad [FOCS'21] of random greedy maximal matching on simple graphs to multigraphs, and prove additional properties that may be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Sublinear Algorithms
  • Steiner Tree
  • Set Cover
  • Maximum Matching
  • Approximation Algorithm

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