Document Open Access Logo

Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs

Authors Syamantak Das , Nikhil Kumar , Daniel Vaz

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2024.45.pdf
  • Filesize: 0.77 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Network flows
  • Theory of computation → Sparsification and spanners
  • Theory of computation → Data compression
Keywords
  • Graph Sparsification
  • Cut Sparsifiers
  • Flow Sparsifiers
  • Quasi-bipartite Graphs
  • Bounded Treewidth

Metrics

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

Abstract

Flow sparsification is a classic graph compression technique which, given a capacitated graph G on k terminals, aims to construct another capacitated graph H, called a flow sparsifier, that preserves, either exactly or approximately, every multicommodity flow between terminals (ideally, with size as a small function of k). Cut sparsifiers are a restricted variant of flow sparsifiers which are only required to preserve maximum flows between bipartitions of the terminal set. It is known that exact cut sparsifiers require 2^Ω(k) many vertices [Krauthgamer and Rika, SODA 2013], with the hard instances being quasi-bipartite graphs, where there are no edges between non-terminals. On the other hand, it has been shown recently that exact (or even (1+ε)-approximate) flow sparsifiers on networks with just 6 terminals require unbounded size [Krauthgamer and Mosenzon, SODA 2023, Chen and Tan, SODA 2024]. 
In this paper, we construct exact flow sparsifiers of size 3^k³ and exact cut sparsifiers of size 2^k² for quasi-bipartite graphs. In particular, the flow sparsifiers are contraction-based, that is, they are obtained from the input graph by (vertex) contraction operations. Our main contribution is a new technique to construct sparsifiers that exploits connections to polyhedral geometry, and that can be generalized to graphs with a small separator that separates the graph into small components. We also give an improved reduction theorem for graphs of bounded treewidth [Andoni et al., SODA 2011], implying a flow sparsifier of size O(k⋅w) and quality O((log w)/log log w), where w is the treewidth.

Cite As Get BibTex

Syamantak Das, Nikhil Kumar, and Daniel Vaz. Nearly-Tight Bounds for Flow Sparsifiers in Quasi-Bipartite Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 45:1-45:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.45

Author Details

Syamantak Das
  • IIIT Delhi, India
Nikhil Kumar
  • University of Waterloo, Canada
Daniel Vaz
  • LAMSADE, CNRS, Université Paris-Dauphine, Université PSL, France

Funding

  • Vaz, Daniel: This work was funded by grants ANR-19-CE48-0016 (Algoridam) and ANR-21-CE48-0022 (S-EX-AP-PE-AL) from the French National Research Agency (ANR), and partially done while the author was at École Normale Superieure and IRIF.

References

  1. Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, and Richard Peng. On fully dynamic graph sparsifiers. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 335-344. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.44.
  2. Alexandr Andoni, Anupam Gupta, and Robert Krauthgamer. Towards (1 + ε)-approximate flow sparsifiers. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 279-293. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.20.
  3. Stefan Arnborg, Derek G. Corneil, and Andrzej Proskurowski. Complexity of finding embeddings in a k-tree. SIAM Journal on Algebraic Discrete Methods, 8(2):277-284, 1987. URL: https://doi.org/10.1137/0608024.
  4. C. A. Barefoot, R. C. Entringer, and H. C. Swart. Vulnerability in graphs - A comparative survey. Journal of Combinatorial Mathematics and Combinatorial Computing, 1, 1987. Google Scholar
  5. András A. Benczúr and David R. Karger. Augmenting undirected edge connectivity in Õ(n^2) time. J. Algorithms, 37(1):2-36, 2000. URL: https://doi.org/10.1006/JAGM.2000.1093.
  6. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  7. Hans L. Bodlaender, Carla Groenland, and Michal Pilipczuk. Parameterized complexity of binary CSP: vertex cover, treedepth, and related parameters. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 27:1-27:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.27.
  8. Gruia Călinescu, Howard J. Karloff, and Yuval Rabani. Approximation algorithms for the 0-extension problem. SIAM J. Comput., 34(2):358-372, 2004. URL: https://doi.org/10.1137/S0097539701395978.
  9. Moses Charikar, Tom Leighton, Shi Li, and Ankur Moitra. Vertex sparsifiers and abstract rounding algorithms. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 265-274. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/FOCS.2010.32.
  10. Shiva Chaudhuri, K. V. Subrahmanyam, Frank Wagner, and Christos D. Zaroliagis. Computing mimicking networks. Algorithmica, 26(1):31-49, 2000. URL: https://doi.org/10.1007/S004539910003.
  11. Yu Chen and Zihan Tan. On (1 + ε)-approximate flow sparsifiers. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 1568-1605. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.63.
  12. Julia Chuzhoy. On vertex sparsifiers with Steiner nodes. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 673-688. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214039.
  13. Michele Conforti, Gerard Cornuejols, and Giacomo Zambelli. Integer Programming. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-11008-0.
  14. Bruno Courcelle. The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  15. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  16. Matthias Englert, Anupam Gupta, Robert Krauthgamer, Harald Räcke, Inbal Talgam-Cohen, and Kunal Talwar. Vertex sparsifiers: New results from old techniques. SIAM J. Comput., 43(4):1239-1262, 2014. URL: https://doi.org/10.1137/130908440.
  17. Jirí Fiala, Petr A. Golovach, and Jan Kratochvíl. Parameterized complexity of coloring problems: Treewidth versus vertex cover. Theor. Comput. Sci., 412(23):2513-2523, 2011. URL: https://doi.org/10.1016/J.TCS.2010.10.043.
  18. Fedor V. Fomin, Mathieu Liedloff, Pedro Montealegre, and Ioan Todinca. Algorithms parameterized by vertex cover and modular width, through potential maximal cliques. Algorithmica, 80(4):1146-1169, 2018. URL: https://doi.org/10.1007/S00453-017-0297-1.
  19. Tatsuya Gima, Tesshu Hanaka, Masashi Kiyomi, Yasuaki Kobayashi, and Yota Otachi. Exploring the gap between treedepth and vertex cover through vertex integrity. Theor. Comput. Sci., 918:60-76, 2022. URL: https://doi.org/10.1016/J.TCS.2022.03.021.
  20. Tatsuya Gima and Yota Otachi. Extended MSO model checking via small vertex integrity. In Sang Won Bae and Heejin Park, editors, 33rd International Symposium on Algorithms and Computation, ISAAC 2022, December 19-21, 2022, Seoul, Korea, volume 248 of LIPIcs, pages 20:1-20:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ISAAC.2022.20.
  21. R. E. Gomory and T. C. Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9(4):551-570, 1961. URL: http://www.jstor.org/stable/2098881.
  22. Torben Hagerup, Jyrki Katajainen, Naomi Nishimura, and Prabhakar Ragde. Characterizing multiterminal flow networks and computing flows in networks of small treewidth. J. Comput. Syst. Sci., 57(3):366-375, 1998. URL: https://doi.org/10.1006/JCSS.1998.1592.
  23. Arun Jambulapati, James R. Lee, Yang P. Liu, and Aaron Sidford. Sparsifying sums of norms. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 1953-1962. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00119.
  24. Arindam Khan and Prasad Raghavendra. On mimicking networks representing minimum terminal cuts. Inf. Process. Lett., 114(7):365-371, 2014. URL: https://doi.org/10.1016/J.IPL.2014.02.011.
  25. Tuukka Korhonen. A single-exponential time 2-approximation algorithm for treewidth. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 184-192. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00026.
  26. Robert Krauthgamer and Ron Mosenzon. Exact flow sparsification requires unbounded size. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 2354-2367. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH91.
  27. Robert Krauthgamer and Inbal Rika. Mimicking networks and succinct representations of terminal cuts. In Sanjeev Khanna, editor, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1789-1799. SIAM, 2013. URL: https://doi.org/10.1137/1.9781611973105.128.
  28. Michael Lampis and Valia Mitsou. Fine-grained meta-theorems for vertex integrity. In Hee-Kap Ahn and Kunihiko Sadakane, editors, 32nd International Symposium on Algorithms and Computation, ISAAC 2021, December 6-8, 2021, Fukuoka, Japan, volume 212 of LIPIcs, pages 34:1-34:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ISAAC.2021.34.
  29. Frank Thomson Leighton and Ankur Moitra. Extensions and limits to vertex sparsification. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 47-56. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806698.
  30. Konstantin Makarychev and Yury Makarychev. Metric extension operators, vertex sparsifiers and lipschitz extendability. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 255-264. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/FOCS.2010.31.
  31. Ankur Moitra. Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 3-12. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/FOCS.2009.28.
  32. Hiroshi Nagamochi and Toshihide Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7(5&6):583-596, 1992. URL: https://doi.org/10.1007/BF01758778.
  33. Jelle J. Oostveen and Erik Jan van Leeuwen. Streaming deletion problems parameterized by vertex cover. Theor. Comput. Sci., 979:114178, 2023. URL: https://doi.org/10.1016/J.TCS.2023.114178.
  34. Sridhar Rajagopalan and Vijay V. Vazirani. On the bidirected cut relaxation for the metric Steiner tree problem. In Robert Endre Tarjan and Tandy J. Warnow, editors, Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, 17-19 January 1999, Baltimore, Maryland, USA, pages 742-751. ACM/SIAM, 1999. URL: http://dl.acm.org/citation.cfm?id=314500.314909.
  35. Daniel A. Spielman and Shang-Hua Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In László Babai, editor, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 81-90. ACM, 2004. URL: https://doi.org/10.1145/1007352.1007372.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact