Document Open Access Logo

Almost-Linear-Time Weighted 𝓁_p-Norm Solvers in Slightly Dense Graphs via Sparsification

Authors Deeksha Adil, Brian Bullins, Rasmus Kyng, Sushant Sachdeva

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2021.9.pdf
  • Filesize: 0.84 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
  • Theory of computation → Network flows
Keywords
  • Weighted 𝓁_p-norm
  • Sparsification
  • Spanners
  • Iterative Refinement

Metrics

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

Abstract

We give almost-linear-time algorithms for constructing sparsifiers with n poly(log n) edges that approximately preserve weighted (𝓁²₂ + 𝓁^p_p) flow or voltage objectives on graphs. For flow objectives, this is the first sparsifier construction for such mixed objectives beyond unit 𝓁_p weights, and is based on expander decompositions. For voltage objectives, we give the first sparsifier construction for these objectives, which we build using graph spanners and leverage score sampling. Together with the iterative refinement framework of [Adil et al, SODA 2019], and a new multiplicative-weights based constant-approximation algorithm for mixed-objective flows or voltages, we show how to find (1+2^{-poly(log n)}) approximations for weighted 𝓁_p-norm minimizing flows or voltages in p(m^{1+o(1)} + n^{4/3 + o(1)}) time for p = ω(1), which is almost-linear for graphs that are slightly dense (m ≥ n^{4/3 + o(1)}).

Cite As Get BibTex

Deeksha Adil, Brian Bullins, Rasmus Kyng, and Sushant Sachdeva. Almost-Linear-Time Weighted 𝓁_p-Norm Solvers in Slightly Dense Graphs via Sparsification. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.9

Author Details

Deeksha Adil
  • University of Toronto, Canada
Brian Bullins
  • Toyota Technological Institute at Chicago, IL, USA
Rasmus Kyng
  • ETH Zurich, Switzerland
Sushant Sachdeva
  • University of Toronto, Canada

Funding

  • Adil, Deeksha: Supported by a Post Graduate Doctoral Scholarship awarded by NSERC (Natural Sciences and Engineering Research Council of Canada) and SS’s Discovery Grant.
  • Sachdeva, Sushant: Supported by a Discovery Grant awarded by NSERC (Natural Sciences and Engineering Research Council of Canada).

References

  1. Deeksha Adil, Rasmus Kyng, Richard Peng, and Sushant Sachdeva. Iterative refinement for 𝓁_p-norm regression. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1405-1424. SIAM, 2019. Google Scholar
  2. Deeksha Adil, Richard Peng, and Sushant Sachdeva. Fast, provably convergent irls algorithm for p-norm linear regression. In Advances in Neural Information Processing Systems, pages 14189-14200, 2019. Google Scholar
  3. Deeksha Adil and Sushant Sachdeva. Faster p-norm minimizing flows, via smoothed q-norm problems. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 892-910. SIAM, 2020. Google Scholar
  4. Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network flows - theory, algorithms and applications. Prentice Hall, 1993. Google Scholar
  5. Zeyuan Allen-Zhu, Yuanzhi Li, Rafael Oliveira, and Avi Wigderson. Much faster algorithms for matrix scaling. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 890-901. IEEE, 2017. Google Scholar
  6. Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9(1):81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
  7. Surender Baswana and Sandeep Sen. A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Struct. Algorithms, 30(4):532–563, 2007. Google Scholar
  8. András A. Benczúr and David R. Karger. Approximating s-t minimum cuts in Õ(n²) time. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, STOC '96, pages 47-55, New York, NY, USA, 1996. ACM. URL: https://doi.org/10.1145/237814.237827.
  9. Sébastien Bubeck, Michael B. Cohen, Yin Tat Lee, and Yuanzhi Li. An homotopy method for lp regression provably beyond self-concordance and in input-sparsity time. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 1130-1137, New York, NY, USA, 2018. ACM. URL: https://doi.org/10.1145/3188745.3188776.
  10. Hui Han Chin, Aleksander Madry, Gary L. Miller, and Richard Peng. Runtime guarantees for regression problems. In Proceedings of the 4superscriptth conference on Innovations in Theoretical Computer Science, ITCS '13, pages 269-282, New York, NY, USA, 2013. ACM. Google Scholar
  11. Paul Christiano, Jonathan A. Kelner, Aleksander Madry, Daniel A. Spielman, and Shang-Hua Teng. Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs. In Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC '11, pages 273-282, New York, NY, USA, 2011. ACM. URL: https://doi.org/10.1145/1993636.1993674.
  12. T. Chu, Y. Gao, R. Peng, S. Sachdeva, S. Sawlani, and J. Wang. Graph sparsification, spectral sketches, and faster resistance computation, via short cycle decompositions. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 361-372, October 2018. URL: https://doi.org/10.1109/FOCS.2018.00042.
  13. M. B. Cohen, A. Madry, D. Tsipras, and A. Vladu. Matrix scaling and balancing via box constrained newton’s method and interior point methods. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 902-913, October 2017. URL: https://doi.org/10.1109/FOCS.2017.88.
  14. Michael B. Cohen, Yin Tat Lee, Cameron Musco, Christopher Musco, Richard Peng, and Aaron Sidford. Uniform sampling for matrix approximation. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS ’15, page 181–190, New York, NY, USA, 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2688073.2688113.
  15. Michael B. Cohen, Yin Tat Lee, and Zhao Song. Solving linear programs in the current matrix multiplication time. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, page 938–942, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3313276.3316303.
  16. Michael B. Cohen, Aleksander Madry, Piotr Sankowski, and Adrian Vladu. Negative-weight shortest paths and unit capacity minimum cost flow in Õ(m^10/7 log w) time (extended abstract). In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 752-771, 2017. Google Scholar
  17. Michael B. Cohen and Richard Peng. 𝓁_p row sampling by lewis weights. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, STOC ’15, page 183–192, New York, NY, USA, 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2746539.2746567.
  18. Samuel I. Daitch and Daniel A. Spielman. Faster approximate lossy generalized flow via interior point algorithms. In Proceedings of the 40th annual ACM symposium on Theory of computing, STOC '08, pages 451-460, New York, NY, USA, 2008. ACM. Available at https://arxiv.org/abs/0803.0988. URL: https://doi.org/10.1145/1374376.1374441.
  19. David Durfee, John Peebles, Richard Peng, and Anup B. Rao. Determinant-preserving sparsification of SDDM matrices with applications to counting and sampling spanning trees. In FOCS, pages 926-937. IEEE Computer Society, 2017. Google Scholar
  20. Andrew V. Goldberg and Robert Endre Tarjan. Efficient maximum flow algorithms. Commun. ACM, 57(8):82-89, 2014. Google Scholar
  21. Tarun Kathuria, Yang P. Liu, and Aaron Sidford. Unit capacity maxflow in almost m^4/3 time. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 119-130, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00020.
  22. Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 217-226, 2014. Google Scholar
  23. Ioannis Koutis, Gary L. Miller, and Richard Peng. A nearly-m log n time solver for SDD linear systems. In Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS '11, pages 590-598, Washington, DC, USA, 2011. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2011.85.
  24. Rasmus Kyng, Yin Tat Lee, Richard Peng, Sushant Sachdeva, and Daniel A Spielman. Sparsified cholesky and multigrid solvers for connection laplacians. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, pages 842-850. ACM, 2016. Google Scholar
  25. Rasmus Kyng, Richard Peng, Sushant Sachdeva, and Di Wang. Flows in almost linear time via adaptive preconditioning. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 902-913, New York, NY, USA, 2019. ACM. URL: https://doi.org/10.1145/3313276.3316410.
  26. Rasmus Kyng, Anup Rao, Sushant Sachdeva, and Daniel A. Spielman. Algorithms for lipschitz learning on graphs. In Peter Grünwald, Elad Hazan, and Satyen Kale, editors, Proceedings of The 28th Conference on Learning Theory, volume 40 of Proceedings of Machine Learning Research, pages 1190-1223, Paris, France, July 2015. PMLR. Google Scholar
  27. Y. T. Lee and A. Sidford. Path finding methods for linear programming: Solving linear programs in Õ(vrank) iterations and faster algorithms for maximum flow. In FOCS, 2014. Google Scholar
  28. Yang P. Liu and Aaron Sidford. Faster energy maximization for faster maximum flow. In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 803-814. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384247.
  29. A. Madry. Navigating central path with electrical flows: From flows to matchings, and back. In FOCS, 2013. Google Scholar
  30. Aleksander Madry. Fast approximation algorithms for cut-based problems in undirected graphs. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 245-254. IEEE, 2010. Google Scholar
  31. Y. Nesterov and A. Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics, 1994. URL: https://doi.org/10.1137/1.9781611970791.
  32. Lorenzo Orecchia, Sushant Sachdeva, and Nisheeth K. Vishnoi. Approximating the exponential, the lanczos method and an Õ(m)-time spectral algorithm for balanced separator. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC '12, pages 1141-1160, New York, NY, USA, 2012. ACM. URL: https://doi.org/10.1145/2213977.2214080.
  33. David Peleg and Alejandro A. Schäffer. Graph spanners. Journal of Graph Theory, 13(1):99-116, 1989. URL: https://doi.org/10.1002/jgt.3190130114.
  34. Richard Peng and Daniel A. Spielman. An efficient parallel solver for SDD linear systems. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC '14, pages 333-342, New York, NY, USA, 2014. ACM. Google Scholar
  35. Harald Racke. Optimal hierarchical decompositions for congestion minimization in networks. In Proceedings of the 40th annual ACM symposium on Theory of computing, STOC '08, pages 255-264, New York, NY, USA, 2008. ACM. Google Scholar
  36. Harald Racke, Chintan Shah, and Hanjo Taubig. Computing cut-based hierarchical decompositions in almost linear time. In Proceedings of the 25superscriptth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA `14, pages 227-238, 2014. Google Scholar
  37. Alexander Schrijver. On the history of the transportation and maximum flow problems. Math. Program., 91(3):437-445, 2002. Google Scholar
  38. Jonah Sherman. Nearly maximum flows in nearly linear time. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 263-269, 2013. Google Scholar
  39. Jonah Sherman. Generalized preconditioning and undirected minimum-cost flow. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 772-780, 2017. Google Scholar
  40. D. Spielman and N. Srivastava. Graph sparsification by effective resistances. SIAM Journal on Computing, 40(6):1913-1926, 2011. URL: https://doi.org/10.1137/080734029.
  41. D. Spielman and S. Teng. Spectral sparsification of graphs. SIAM Journal on Computing, 40(4):981-1025, 2011. URL: https://doi.org/10.1137/08074489X.
  42. D.A. Spielman and S. Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In STOC, 2004. Google Scholar
  43. Daniel A. Spielman. Spectral graph theory lectures: Sparsification by effective resistance sampling, 2015. Google Scholar
  44. Jan van den Brand, Yin-Tat Lee, Danupon Nanongkai, Richard Peng, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, and Di Wang. Bipartite matching in nearly-linear time on moderately dense graphs. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 919-930. IEEE, 2020. Google Scholar
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact