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Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary

Authors Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg, Danupon Nanongkai, Thatchaphol Saranurak, Aaron Sidford, He Sun

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Author Details

Aaron Bernstein
  • Rutgers University, Piscataway, NJ, USA
Jan van den Brand
  • Simons Institute, Berkeley, CA, USA
  • University of California Berkeley, CA, USA
Maximilian Probst Gutenberg
  • ETH Zürich, Switzerland
Danupon Nanongkai
  • University of Copenhagen, Denmark
  • KTH Royal Institute of Technology, Stockholm, Sweden
Thatchaphol Saranurak
  • University of Michigan, Ann Arbor, MI, USA
Aaron Sidford
  • Stanford University, CA, USA
He Sun
  • University of Edinburgh, UK

Funding

  • Bernstein, Aaron: Funded by NSF Career grant 1942010.
  • van den Brand, Jan: Funded by ONR BRC grant N00014-18-1-2562 and by the Simons Institute for the Theory of Computing through a Simons-Berkeley Postdoctoral Fellowship. Research partially done at KTH.
  • Probst Gutenberg, Maximilian: Research partially done while at University of Copenhagen.
  • Nanongkai, Danupon: This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No 715672. Also partially supported by the Swedish Research Council (Reg. No. 2015-04659 and 2019-05622).
  • Saranurak, Thatchaphol: Research partially done at KTH and supported by the Swedish Research Council (Reg. No. 2015-04659).
  • Sidford, Aaron: Supported in part by a Microsoft Research Faculty Fellowship, NSF CAREER Award CCF-1844855, NSF Grant CCF-1955039, a PayPal research award, and a Sloan Research Fellowship.
  • Sun, He: Funded by an EPSRC Early Career Fellowship (EP/T00729X/1).

Acknowledgements

We thank Julia Chuzhoy and Gramoz Goranci for discussions.

Cite As Get BibTex

Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg, Danupon Nanongkai, Thatchaphol Saranurak, Aaron Sidford, and He Sun. Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.20

Abstract

Designing efficient dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms and has witnessed many exciting recent developments in, e.g., dynamic matching (Wajc STOC'20) and decremental shortest paths (Chuzhoy and Khanna STOC'19). Compared to other graph primitives (e.g. spanning trees and matchings), designing such algorithms for graph spanners and (more broadly) graph sparsifiers poses a unique challenge since there is no fast deterministic algorithm known for static computation and the lack of a way to adjust the output slowly (known as "small recourse/replacements"). 
This paper presents the first non-trivial efficient adaptive algorithms for maintaining many sparsifiers against an adaptive adversary. Specifically, we present algorithms that maintain  
1) a polylog(n)-spanner of size Õ(n) in polylog(n) amortized update time, 
2) an O(k)-approximate cut sparsifier of size Õ(n) in Õ(n^{1/k}) amortized update time, and 
3) a polylog(n)-approximate spectral sparsifier in polylog(n) amortized update time.  Our bounds are the first non-trivial ones even when only the recourse is concerned. Our results hold even against a stronger adversary, who can access the random bits previously used by the algorithms and the amortized update time of all algorithms can be made worst-case by paying sub-polynomial factors. Our spanner result resolves an open question by Ahmed et al. (2019) and our results and techniques imply additional improvements over existing results, including (i) answering open questions about decremental single-source shortest paths by Chuzhoy and Khanna (STOC'19) and Gutenberg and Wulff-Nilsen (SODA'20), implying a nearly-quadratic time algorithm for approximating minimum-cost unit-capacity flow and (ii) de-amortizing a result of Abraham et al. (FOCS'16) for dynamic spectral sparsifiers.
Our results are based on two novel techniques. The first technique is a generic black-box reduction that allows us to assume that the graph is initially an expander with almost uniform-degree and, more importantly, stays as an almost uniform-degree expander while undergoing only edge deletions. The second technique is called proactive resampling: here we constantly re-sample parts of the input graph so that, independent of an adversary’s computational power, a desired structure of the underlying graph can be always maintained. Despite its simplicity, the analysis of this sampling scheme is far from trivial, because the adversary can potentially create dependencies between the random choices used by the algorithm. We believe these two techniques could be useful for developing other adaptive algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • dynamic graph algorithm
  • adaptive adversary
  • spanner
  • sparsifier

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