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Space Optimal Vertex Cover in Dynamic Streams

Authors Kheeran K. Naidu , Vihan Shah

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Author Details

Kheeran K. Naidu
  • Department of Computer Science, University of Bristol, UK
Vihan Shah
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA

Funding

  • Naidu, Kheeran K.: EPSRC Doctoral Training Studentship EP/T517872/1.
  • Shah, Vihan: Research supported in part by a NSF CAREER Grant CCF-2047061.

Acknowledgements

We are grateful to Sepehr Assadi and Christian Konrad for many helpful discussions. We also appreciate the valuable comments from our APPROX 2022 reviewers.

Cite As Get BibTex

Kheeran K. Naidu and Vihan Shah. Space Optimal Vertex Cover in Dynamic Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.53

Abstract

We optimally resolve the space complexity for the problem of finding an α-approximate minimum vertex cover (αMVC) in dynamic graph streams. We give a randomised algorithm for αMVC which uses O(n²/α²) bits of space matching Dark and Konrad’s lower bound [CCC 2020] up to constant factors. By computing a random greedy matching, we identify "easy" instances of the problem which can trivially be solved by returning the entire vertex set. The remaining "hard" instances, then have sparse induced subgraphs which we exploit to get our space savings and solve αMVC.
Achieving this type of optimality result is crucial for providing a complete understanding of a problem, and it has been gaining interest within the dynamic graph streaming community. For connectivity, Nelson and Yu [SODA 2019] improved the lower bound showing that Ω(n log³ n) bits of space is necessary while Ahn, Guha, and McGregor [SODA 2012] have shown that O(n log³ n) bits is sufficient. For finding an α-approximate maximum matching, the upper bound was improved by Assadi and Shah [ITCS 2022] showing that O(n²/α³) bits is sufficient while Dark and Konrad [CCC 2020] have shown that Ω(n²/α³) bits is necessary. The space complexity, however, remains unresolved for many other dynamic graph streaming problems where further improvements can still be made.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph Streaming Algorithms
  • Vertex Cover
  • Dynamic Streams
  • Approximation Algorithm

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