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Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy

Authors Sepehr Assadi , Prantar Ghosh , Bruno Loff , Parth Mittal , Sagnik Mukhopadhyay

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Author Details

Sepehr Assadi
  • Cheriton School of Computer Science, University of Waterloo, Canada
Prantar Ghosh
  • Department of Computer Science, Georgetown University, Washington, DC, USA
Bruno Loff
  • Department of Mathematics and LASIGE, Faculdade de Ciências, Universidade de Lisboa, Portugal
Parth Mittal
  • University of Waterloo, Canada
Sagnik Mukhopadhyay
  • University of Sheffield, UK

Funding

  • Assadi, Sepehr: Supported in part by a Sloan Research Fellowship, an NSERC Discovery Grant, a University of Waterloo startup grant, and a Faculty of Math Research Chair grant.
  • Ghosh, Prantar: Supported in part by NSF under award 1918989. Part of this work was done while he was at DIMACS, Rutgers University, supported in part by a grant (820931) to DIMACS from the Simons Foundation.
  • Loff, Bruno: Funded by the European Union (ERC, HOFGA, 101041696). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. He was also supported by FCT through the LASIGE Research Unit, ref. UIDB/00408/2020 and ref. UIDP/00408/2020, and by CMAFcIO, FCT Project UIDB/04561/2020, https://doi.org/10.54499/UIDB/04561/2020.
  • Mittal, Parth: Supported by a David R. Cheriton Graduate Scholarship and Sepehr Assadi’s Sloan Research Fellowship, an NSERC Discovery grant, and a startup grant from University of Waterloo.
  • Mukhopadhyay, Sagnik: Partially funded by the UKRI New Investigator Award, ref. https://gow.epsrc.ukri.org/NGBOViewGrant.aspx?GrantRef=EP/X03805X/1.

Acknowledgements

The first named author would like to thank Yu Chen and Sanjeev Khanna for their collaboration in [Sepehr Assadi et al., 2019] that was the starting point of this project and Madhu Sudan for helpful conversations.

Cite AsGet BibTex

Sepehr Assadi, Prantar Ghosh, Bruno Loff, Parth Mittal, and Sagnik Mukhopadhyay. Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.7

Abstract

The following question arises naturally in the study of graph streaming algorithms: Is there any graph problem which is "not too hard", in that it can be solved efficiently with total communication (nearly) linear in the number n of vertices, and for which, nonetheless, any streaming algorithm with Õ(n) space (i.e., a semi-streaming algorithm) needs a polynomial n^Ω(1) number of passes? Assadi, Chen, and Khanna [STOC 2019] were the first to prove that this is indeed the case. However, the lower bounds that they obtained are for rather non-standard graph problems. Our first main contribution is to present the first polynomial-pass lower bounds for natural "not too hard" graph problems studied previously in the streaming model: k-cores and degeneracy. We devise a novel communication protocol for both problems with near-linear communication, thus showing that k-cores and degeneracy are natural examples of "not too hard" problems. Indeed, previous work have developed single-pass semi-streaming algorithms for approximating these problems. In contrast, we prove that any semi-streaming algorithm for exactly solving these problems requires (almost) Ω(n^{1/3}) passes. The lower bound follows by a reduction from a generalization of the hidden pointer chasing (HPC) problem of Assadi, Chen, and Khanna, which is also the basis of their earlier semi-streaming lower bounds. Our second main contribution is improved round-communication lower bounds for the underlying communication problems at the basis of these reductions: - We improve the previous lower bound of Assadi, Chen, and Khanna for HPC to achieve optimal bounds for this problem. - We further observe that all current reductions from HPC can also work with a generalized version of this problem that we call MultiHPC, and prove an even stronger and optimal lower bound for this generalization. These two results collectively allow us to improve the resulting pass lower bounds for semi-streaming algorithms by a polynomial factor, namely, from n^{1/5} to n^{1/3} passes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Lower bounds and information complexity
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph streaming
  • Lower bounds
  • Communication complexity
  • k-Cores and degeneracy

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