Document Open Access Logo

Parameterized Complexity of Streaming Diameter and Connectivity Problems

Authors Jelle J. Oostveen, Erik Jan van Leeuwen

PDF
Thumbnail PDF

File

LIPIcs.IPEC.2022.24.pdf
  • Filesize: 0.98 MB
  • 16 pages

Document Identifiers

Author Details

Jelle J. Oostveen
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Erik Jan van Leeuwen
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Funding

  • Oostveen, Jelle J.: This author is partially supported by the NWO grant OCENW.KLEIN. 114 (PACAN).

Acknowledgements

We thank the reviewers for their helpful comments and feedback.

Cite AsGet BibTex

Jelle J. Oostveen and Erik Jan van Leeuwen. Parameterized Complexity of Streaming Diameter and Connectivity Problems. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.IPEC.2022.24

Abstract

We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size k allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is 𝒪(log n) for any fixed k. Underlying these algorithms is a method to execute a breadth-first search in 𝒪(k) passes and 𝒪(k log n) bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where Ω(n/p) bits of memory is needed for any p-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph H, for most H. For some cases, we can also show one-pass, Ω(n log n) bits of memory lower bounds. We also prove a much stronger Ω(n²/p) lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size k. This yields a kernel of 2k vertices (with 𝒪(k²) edges) produced as a stream in poly(k) passes and only 𝒪(k log n) bits of memory.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Lower bounds and information complexity
Keywords
  • Stream
  • Streaming
  • Graphs
  • Parameter
  • Complexity
  • Diameter
  • Connectivity
  • Vertex Cover
  • Disjointness
  • Permutation

Metrics

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

Related Versions

References

  1. Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Proc. SODA 2016, pages 377-391. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch28.
  2. Deepak Agarwal, Andrew McGregor, Jeff M. Phillips, Suresh Venkatasubramanian, and Zhengyuan Zhu. Spatial scan statistics: approximations and performance study. In Proc. SIGKDD 2006, pages 24-33. ACM, 2006. URL: https://doi.org/10.1145/1150402.1150410.
  3. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Polynomial pass lower bounds for graph streaming algorithms. CoRR, abs/1904.04720, 2019. URL: http://arxiv.org/abs/1904.04720.
  4. Sepehr Assadi, Gillat Kol, Raghuvansh R. Saxena, and Huacheng Yu. Multi-pass graph streaming lower bounds for cycle counting, max-cut, matching size, and other problems. In Proc. FOCS 2020, pages 354-364. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00041.
  5. Sepehr Assadi and Vishvajeet N. Graph streaming lower bounds for parameter estimation and property testing via a streaming XOR lemma. In Proc. STOC 2021, pages 612-625. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451110.
  6. Sepehr Assadi and Ran Raz. Near-quadratic lower bounds for two-pass graph streaming algorithms. In Proc. FOCS 2020, pages 342-353. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00040.
  7. Reuven Bar-Yehuda and Shimon Even. A local-ratio theorem for approximating the weighted vertex cover problem. In Proc. WG 1983, pages 17-28. Universitätsverlag Rudolf Trauner, Linz, 1983. URL: http://www.gbv.de/dms/tib-ub-hannover/022054669.pdf.
  8. Matthias Bentert and André Nichterlein. Parameterized complexity of diameter. In Proc. CIAC 2019, volume 11485 of LNCS, pages 50-61. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-17402-6_5.
  9. Arijit Bishnu, Arijit Ghosh, Sudeshna Kolay, Gopinath Mishra, and Saket Saurabh. Fixed-parameter tractability of graph deletion problems over data streams. CoRR, abs/1906.05458, 2019. URL: http://arxiv.org/abs/1906.05458.
  10. Arijit Bishnu, Arijit Ghosh, Sudeshna Kolay, Gopinath Mishra, and Saket Saurabh. Fixed parameter tractability of graph deletion problems over data streams. In Proc. COCOON 2020, volume 12273 of LNCS, pages 652-663. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-58150-3_53.
  11. Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, and Sandeep Sen. On the streaming complexity of fundamental geometric problems. CoRR, abs/1803.06875, 2018. URL: http://arxiv.org/abs/1803.06875.
  12. J. Adrian Bondy and Uppaluri S. R. Murty. Graph Theory with Applications. Macmillan Education UK, 1976. URL: https://doi.org/10.1007/978-1-349-03521-2.
  13. Karl Bringmann, Thore Husfeldt, and Måns Magnusson. Multivariate analysis of orthogonal range searching and graph distances. Algorithmica, 82(8):2292-2315, 2020. URL: https://doi.org/10.1007/s00453-020-00680-z.
  14. Jonathan F. Buss and Judy Goldsmith. Nondeterminism within P. SIAM J. Comput., 22(3):560-572, 1993. URL: https://doi.org/10.1137/0222038.
  15. Sergio Cabello. Subquadratic algorithms for the diameter and the sum of pairwise distances in planar graphs. ACM Trans. Algorithms, 15(2):21:1-21:38, 2019. URL: https://doi.org/10.1145/3218821.
  16. Jianer Chen, Iyad A. Kanj, and Weijia Jia. Vertex cover: Further observations and further improvements. Journal of Algorithms, 41(2):280-301, 2001. URL: https://doi.org/10.1006/jagm.2001.1186.
  17. Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, and Huacheng Yu. Almost optimal super-constant-pass streaming lower bounds for reachability. In Proc. STOC 2021, pages 570-583. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451038.
  18. Rajesh Chitnis and Graham Cormode. Towards a theory of parameterized streaming algorithms. In Proc. IPEC 2019, volume 148 of LIPIcs, pages 7:1-7:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.IPEC.2019.7.
  19. Rajesh Chitnis, Graham Cormode, Hossein Esfandiari, Mohammad Taghi Hajiaghayi, Andrew McGregor, Morteza Monemizadeh, and Sofya Vorotnikova. Kernelization via sampling with applications to finding matchings and related problems in dynamic graph streams. In Proc. SODA 2016, pages 1326-1344. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch92.
  20. Rajesh Hemant Chitnis, Graham Cormode, Hossein Esfandiari, MohammadTaghi Hajiaghayi, and Morteza Monemizadeh. Brief announcement: New streaming algorithms for parameterized maximal matching & beyond. In Proc. SPAA 2015, pages 56-58. ACM, 2015. URL: https://doi.org/10.1145/2755573.2755618.
  21. Rajesh Hemant Chitnis, Graham Cormode, Mohammad Taghi Hajiaghayi, and Morteza Monemizadeh. Parameterized streaming: Maximal matching and vertex cover. In Proc. SODA 2015, pages 1234-1251. SIAM, 2015. URL: https://doi.org/10.1137/1.9781611973730.82.
  22. Derek G. Corneil, Feodor F. Dragan, Michel Habib, and Christophe Paul. Diameter determination on restricted graph families. Discret. Appl. Math., 113(2-3):143-166, 2001. URL: https://doi.org/10.1016/S0166-218X(00)00281-X.
  23. David Coudert, Guillaume Ducoffe, and Alexandru Popa. Fully polynomial FPT algorithms for some classes of bounded clique-width graphs. ACM Trans. Algorithms, 15(3):33:1-33:57, 2019. URL: https://doi.org/10.1145/3310228.
  24. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL: https://doi.org/10.1145/2629620.
  25. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999. URL: https://doi.org/10.1007/978-1-4612-0515-9.
  26. Guillaume Ducoffe. Beyond helly graphs: The diameter problem on absolute retracts. In Proc. WG 2021, volume 12911 of LNCS, pages 321-335. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-86838-3_25.
  27. Guillaume Ducoffe and Feodor F. Dragan. A story of diameter, radius, and (almost) helly property. Networks, 77(3):435-453, 2021. URL: https://doi.org/10.1002/net.21998.
  28. Guillaume Ducoffe, Michel Habib, and Laurent Viennot. Fast diameter computation within split graphs. In Proc. COCOA 2019, volume 11949 of LNCS, pages 155-167. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-36412-0_13.
  29. Guillaume Ducoffe, Michel Habib, and Laurent Viennot. Diameter computation on H-minor free graphs and graphs of bounded (distance) vc-dimension. In Proc. SODA 2020, pages 1905-1922. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.117.
  30. Michael Elkin. Distributed exact shortest paths in sublinear time. J. ACM, 67(3):15:1-15:36, 2020. URL: https://doi.org/10.1145/3387161.
  31. Michael Elkin and Chhaya Trehan. (1+ε)-approximate shortest paths in dynamic streams. CoRR, abs/2107.13309, 2021. URL: http://arxiv.org/abs/2107.13309.
  32. Stefan Fafianie and Stefan Kratsch. Streaming kernelization. In Proc. MFCS 2014, volume 8635 of LNCS, pages 275-286. Springer, 2014. Google Scholar
  33. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theor. Comput. Sci., 348(2-3):207-216, 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.013.
  34. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. Graph distances in the data-stream model. SIAM J. Comput., 38(5):1709-1727, 2008. URL: https://doi.org/10.1137/070683155.
  35. Pawel Gawrychowski, Haim Kaplan, Shay Mozes, Micha Sharir, and Oren Weimann. Voronoi diagrams on planar graphs, and computing the diameter in deterministic õ(n^5/3) time. SIAM J. Comput., 50(2):509-554, 2021. URL: https://doi.org/10.1137/18M1193402.
  36. Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Yuval Rabani, editor, Proc. SODA 2012, pages 468-485. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.41.
  37. Venkatesan Guruswami and Krzysztof Onak. Superlinear lower bounds for multipass graph processing. Algorithmica, 76(3):654-683, 2016. URL: https://doi.org/10.1007/s00453-016-0138-7.
  38. Monika Rauch Henzinger, Prabhakar Raghavan, and Sridhar Rajagopalan. Computing on data streams. In Proc. DIMACS 1998, volume 50 of DIMACS, pages 107-118. DIMACS/AMS, 1998. URL: https://doi.org/10.1090/dimacs/050/05.
  39. Zengfeng Huang and Pan Peng. Dynamic graph stream algorithms in o(n) space. Algorithmica, 81(5):1965-1987, 2019. URL: https://doi.org/10.1007/s00453-018-0520-8.
  40. Thore Husfeldt. Computing graph distances parameterized by treewidth and diameter. In Proc. IPEC 2016, volume 63 of LIPIcs, pages 16:1-16:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.IPEC.2016.16.
  41. Michael Kapralov. Better bounds for matchings in the streaming model. In Sanjeev Khanna, editor, Proc. SODA 2013, pages 1679-1697. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.121.
  42. Shahbaz Khan and Shashank K. Mehta. Depth first search in the semi-streaming model. In Rolf Niedermeier and Christophe Paul, editors, Proc. STACS 2019, volume 126 of LIPIcs, pages 42:1-42:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.STACS.2019.42.
  43. Andrew McGregor. Graph stream algorithms: a survey. SIGMOD Rec., 43(1):9-20, 2014. URL: https://doi.org/10.1145/2627692.2627694.
  44. Andrew McGregor, Sofya Vorotnikova, and Hoa T. Vu. Better algorithms for counting triangles in data streams. In Tova Milo and Wang-Chiew Tan, editors, Proc. PODS 2016, pages 401-411. ACM, 2016. URL: https://doi.org/10.1145/2902251.2902283.
  45. George L. Nemhauser and Leslie E. Trotter Jr. Vertex packings: Structural properties and algorithms. Math. Program., 8(1):232-248, 1975. URL: https://doi.org/10.1007/BF01580444.
  46. Jelle J. Oostveen and Erik Jan van Leeuwen. Streaming deletion problems parameterized by vertex cover. In Proc. FCT 2021, volume 12867 of LNCS, pages 413-426. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-86593-1_29.
  47. Jelle J. Oostveen and Erik Jan van Leeuwen. Parameterized complexity of streaming diameter and connectivity problems. CoRR, abs/2207.04872, 2022. URL: https://doi.org/10.48550/arXiv.2207.04872.
  48. John H. Reif. Depth-first search is inherently sequential. Inf. Process. Lett., 20(5):229-234, 1985. URL: https://doi.org/10.1016/0020-0190(85)90024-9.
  49. Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proc. STOC 2013, pages 515-524. ACM, 2013. URL: https://doi.org/10.1145/2488608.2488673.
  50. Xiaoming Sun and David P. Woodruff. Tight bounds for graph problems in insertion streams. In Proc. APPROX/RANDOM 2015, volume 40 of LIPIcs, pages 435-448. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.435.
  51. Elad Verbin and Wei Yu. The streaming complexity of cycle counting, sorting by reversals, and other problems. In Dana Randall, editor, Proc. SODA 2011, pages 11-25. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.2.
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