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Streaming Complexity Separations for Dense and Sparse Graphs

Authors Yang P. Liu , Hoai-An Nguyen , Noah G. Singer , David P. Woodruff

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ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • streaming
  • maximum cut
  • complexity separations

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Abstract

We identify a sharp separation in the streaming space complexity of Maximum Cut when the algorithm must output an approximate cut (rather than only the approximate value). For dense graphs, we show that O(n/ε²) space is sufficient and that Ω(n) space is necessary. In contrast, for graphs with Θ(n/ε²) edges, the situation is markedly different: we show that the problem requires Ω(n log(ε² n)/ε²) space for any ε = ω(1/√n), which is tight for the full range of ε. We also give an Ω(n log n/ε²)-space lower bound against deterministic algorithms for outputting a (1-ε) approximation to the value of the maximum cut.
Using similar techniques we prove an analogous sharp separation in the streaming space complexity of Densest Subgraph and show that for every constant-arity CSP over a constant-size alphabet and the Similarity problem the space complexity in dense streams can be improved by shaving a logarithmic factor.

Cite As Get BibTex

Yang P. Liu, Hoai-An Nguyen, Noah G. Singer, and David P. Woodruff. Streaming Complexity Separations for Dense and Sparse Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 140:1-140:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.140

Author Details

Yang P. Liu
  • Carnegie Mellon University, Pittsburgh, PA, USA
Hoai-An Nguyen
  • Carnegie Mellon University, Pittsburgh, PA, USA
Noah G. Singer
  • Carnegie Mellon University, Pittsburgh, PA, USA
David P. Woodruff
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Nguyen, Hoai-An: Supported in part by NSF GRFP grant number DGE2140739, Office of Naval Research award number N000142112647, and a Simons Investigator Award.
  • Woodruff, David P.: Supported in part by Office of Naval Research award number N000142112647 and a Simons Investigator Award.

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