Document Open Access Logo

Lower Bounds for Approximating Graph Parameters via Communication Complexity

Authors Talya Eden, Will Rosenbaum

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2018.11.pdf
  • Filesize: 0.54 MB
  • 18 pages

Document Identifiers

Author Details

Talya Eden
  • School of Electrical Engineering, Tel Aviv University, Tel Aviv 6997801, Israel
Will Rosenbaum
  • School of Electrical Engineering, Tel Aviv University, Tel Aviv 6997801, Israel

Funding

  • Eden, Talya: This research was partially supported by a grant from the Blavatnik fund. The author is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship, and for the Weinstein Institute for their support.

Cite AsGet BibTex

Talya Eden and Will Rosenbaum. Lower Bounds for Approximating Graph Parameters via Communication Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.11

Abstract

In a celebrated work, Blais, Brody, and Matulef [Blais et al., 2012] developed a technique for proving property testing lower bounds via reductions from communication complexity. Their work focused on testing properties of functions, and yielded new lower bounds as well as simplified analyses of known lower bounds. Here, we take a further step in generalizing the methodology of [Blais et al., 2012] to analyze the query complexity of graph parameter estimation problems. In particular, our technique decouples the lower bound arguments from the representation of the graph, allowing it to work with any query type. We illustrate our technique by providing new simpler proofs of previously known tight lower bounds for the query complexity of several graph problems: estimating the number of edges in a graph, sampling edges from an almost-uniform distribution, estimating the number of triangles (and more generally, r-cliques) in a graph, and estimating the moments of the degree distribution of a graph. We also prove new lower bounds for estimating the edge connectivity of a graph and estimating the number of instances of any fixed subgraph in a graph. We show that the lower bounds for estimating the number of triangles and edge connectivity also hold in a strictly stronger computational model that allows access to uniformly random edge samples.

Subject Classification

ACM Subject Classification
  • Theory of computation → Lower bounds and information complexity
Keywords
  • sublinear graph parameter estimation
  • lower bounds
  • communication complexity

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Martin Aigner and Günter M Ziegler. Proofs from the Book. Springer-Verlag Berlin Heidelberg, 4th edition, 2010. URL: http://dx.doi.org/10.1007/978-3-642-00856-6.
  2. Maryam Aliakbarpour, Amartya Shankha Biswas, Themistoklis Gouleakis, John Peebles, Ronitt Rubinfeld, and Anak Yodpinyanee. Sublinear-time algorithms for counting star subgraphs with applications to join selectivity estimation. CoRR, abs/1601.04233, 2016. URL: http://arxiv.org/abs/1601.04233.
  3. Noga Alon, Tali Kaufman, Michael Krivelevich, and Dana Ron. Testing triangle-freeness in general graphs. SIAM Journal on Discrete Mathematics, 22(2):786-819, 2008. Google Scholar
  4. Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 20-29. ACM, 1996. Google Scholar
  5. Ziv Bar-Yossef, T.S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. Journal of Computer and System Sciences, 68(4):702-732, 2004. Special Issue on FOCS 2002. URL: http://dx.doi.org/10.1016/j.jcss.2003.11.006.
  6. Suman K. Bera and Amit Chakrabarti. Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams. In Heribert Vollmer and Brigitte Vallée, editors, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), volume 66 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:14, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2017.11.
  7. Eric Blais, Joshua Brody, and Kevin Matulef. Property testing lower bounds via communication complexity. computational complexity, 21(2):311-358, Jun 2012. URL: http://dx.doi.org/10.1007/s00037-012-0040-x.
  8. Shahar Dobzinski, Noam Nisan, and Michael Schapira. Approximation algorithms for combinatorial auctions with complement-free bidders. In Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing, STOC '05, pages 610-618, New York, NY, USA, 2005. ACM. URL: http://dx.doi.org/10.1145/1060590.1060681.
  9. T. Eden, A. Levi, D. Ron, and C. Seshadhri. Approximately counting triangles in sublinear time. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 614-633, Oct 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.44.
  10. Talya Eden, Dana Ron, and C. Seshadhri. On approximating the number of k-cliques in sublinear time. CoRR, abs/1707.04858, 2017. URL: http://arxiv.org/abs/1707.04858.
  11. Talya Eden, Dana Ron, and C. Seshadhri. Sublinear time estimation of degree distribution moments: The degeneracy connection. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 7:1-7:13, 2017. Full version available at https://arxiv.org/abs/1604.03661. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.7.
  12. Talya Eden and Will Rosenbaum. Lower bounds for approximating graph parameters via communication complexity. Full version of this paper, 2017. URL: https://arxiv.org/abs/1709.04262.
  13. Talya Eden and Will Rosenbaum. On Sampling Edges Almost Uniformly. In Raimund Seidel, editor, 1st Symposium on Simplicity in Algorithms (SOSA 2018), volume 61 of OpenAccess Series in Informatics (OASIcs), pages 7:1-7:9, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Full version available at https://arxiv.org/abs/1706.09748. URL: http://dx.doi.org/10.4230/OASIcs.SOSA.2018.7.
  14. Uri Feige. On sums of independent random variables with unbounded variance and estimating the average degree in a graph. SIAM J. Comput., 35(4):964-984, 2006. URL: http://dx.doi.org/10.1137/S0097539704447304.
  15. Oded Goldreich. On the communication complexity methodology for proving lower bounds on the query complexity of property testing. Electronic Colloquium on Computational Complexity (ECCC), 20:73, 2013. URL: http://eccc.hpi-web.de/report/2013/073.
  16. Oded Goldreich, Shari Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM (JACM), 45(4):653-750, 1998. Google Scholar
  17. Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. Algorithmica, pages 302-343, 2002. Google Scholar
  18. Oded Goldreich and Dana Ron. Approximating average parameters of graphs. Random Structures &Algorithms, 32(4):473-493, 2008. URL: http://dx.doi.org/10.1002/rsa.20203.
  19. Yannai A. Gonczarowski, Noam Nisan, Rafail Ostrovsky, and Will Rosenbaum. A stable marriage requires communication. In Proceedings of the Twenty-sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pages 1003-1017, Philadelphia, PA, USA, 2015. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2722129.2722197.
  20. Mira Gonen, Dana Ron, and Yuval Shavitt. Counting stars and other small subgraphs in sublinear-time. SIAM Journal on Discrete Mathematics, 25(3):1365-1411, 2011. Google Scholar
  21. J. Hromkovič. Communication Complexity and Parallel Computing. Texts in Theoretical Computer Science. An EATCS Series. Springer Berlin Heidelberg, 2013. Google Scholar
  22. Bala Kalyanasundaram and Georg Schintger. The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics, 5(4):545-557, 1992. Google Scholar
  23. Bala Kalyanasundaram and Georg Schnitger. Communication Complexity and Lower Bounds for Sequential Computation, pages 253-268. Vieweg+Teubner Verlag, Wiesbaden, 1992. URL: http://dx.doi.org/10.1007/978-3-322-95233-2_15.
  24. Mauricio Karchmer and Avi Wigderson. Monotone circuits for connectivity require super-logarithmic depth. SIAM Journal on Discrete Mathematics, 3(2):255-265, 1990. URL: http://dx.doi.org/10.1137/0403021.
  25. Tali Kaufman, Michael Krivelevich, and Dana Ron. Tight bounds for testing bipartiteness in general graphs. SIAM Journal on computing, 33(6):1441-1483, 2004. Google Scholar
  26. E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, 2006. Google Scholar
  27. Rajeev Motwani, Rina Panigrahy, and Ying Xu. Estimating sum by weighted sampling. In ICALP, volume 4596, pages 53-64. Springer, 2007. Google Scholar
  28. Michal Parnas and Dana Ron. Testing the diameter of graphs. Random Structures &Algorithms, 20(2):165-183, 2002. URL: http://dx.doi.org/10.1002/rsa.10013.
  29. Ramamohan Paturi and Janos Simon. Probabilistic communication complexity. Journal of Computer and System Sciences, 33(1):106-123, 1986. Google Scholar
  30. Alexander A. Razborov. On the distributional complexity of disjointness. Theoretical Computer Science, 106(2):385-390, 1992. Google Scholar
  31. Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, and Roger Wattenhofer. Distributed verification and hardness of distributed approximation. SIAM Journal on Computing, 41(5):1235-1265, 2012. URL: http://dx.doi.org/10.1137/11085178X.
  32. Jacob Steinhardt, Gregory Valiant, and Stefan Wager. Memory, communication, and statistical queries. In Vitaly Feldman, Alexander Rakhlin, and Ohad Shamir, editors, 29th Annual Conference on Learning Theory, volume 49 of Proceedings of Machine Learning Research, pages 1490-1516, Columbia University, New York, New York, USA, 23-26 Jun 2016. PMLR. URL: http://proceedings.mlr.press/v49/steinhardt16.html.
  33. Paul Turán. On an extremal problem in graph theory. Matematikai és Fizikai Lapok, 48:436-452, 1941. Google Scholar
  34. Andrew Chi-Chih Yao. Some complexity questions related to distributive computing(preliminary report). In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC '79, pages 209-213, New York, NY, USA, 1979. ACM. URL: http://dx.doi.org/10.1145/800135.804414.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact