Better Sparsifiers for Directed Eulerian Graphs

Authors Sushant Sachdeva , Anvith Thudi , Yibin Zhao



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Sushant Sachdeva
  • University of Toronto, Canada
Anvith Thudi
  • University of Toronto, Canada
Yibin Zhao
  • University of Toronto, Canada

Acknowledgements

We thank Arun Jambulapati for notifying us of an issue in a previous version of this manuscript. SS and YZ would also like to thank the Simons Institute for the Theory of Computing Fall 2023 program for its support and where a significant part of this project evolved.

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Sushant Sachdeva, Anvith Thudi, and Yibin Zhao. Better Sparsifiers for Directed Eulerian Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 119:1-119:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.119

Abstract

Spectral sparsification for directed Eulerian graphs is a key component in the design of fast algorithms for solving directed Laplacian linear systems. Directed Laplacian linear system solvers are crucial algorithmic primitives to fast computation of fundamental problems on random walks, such as computing stationary distributions, hitting and commute times, and personalized PageRank vectors. While spectral sparsification is well understood for undirected graphs and it is known that for every graph G, (1+ε)-sparsifiers with O(nε^{-2}) edges exist [Batson-Spielman-Srivastava, STOC '09] (which is optimal), the best known constructions of Eulerian sparsifiers require Ω(nε^{-2}log⁴ n) edges and are based on short-cycle decompositions [Chu et al., FOCS '18]. In this paper, we give improved constructions of Eulerian sparsifiers, specifically: 1) We show that for every directed Eulerian graph G→, there exists an Eulerian sparsifier with O(nε^{-2} log² n log²log n + nε^{-4/3}log^{8/3} n) edges. This result is based on combining short-cycle decompositions [Chu-Gao-Peng-Sachdeva-Sawlani-Wang, FOCS '18, SICOMP] and [Parter-Yogev, ICALP '19], with recent progress on the matrix Spencer conjecture [Bansal-Meka-Jiang, STOC '23]. 2) We give an improved analysis of the constructions based on short-cycle decompositions, giving an m^{1+δ}-time algorithm for any constant δ > 0 for constructing Eulerian sparsifiers with O(nε^{-2}log³ n) edges.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Linear algebra algorithms
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Computations on matrices
Keywords
  • Graph algorithms
  • Linear algebra and computation
  • Discrepancy theory

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References

  1. AmirMahdi Ahmadinejad, Arun Jambulapati, Amin Saberi, and Aaron Sidford. Perron-Frobenius Theory in Nearly Linear Time: Positive Eigenvectors, M-matrices, Graph Kernels, and Other Applications, pages 1387-1404. Society for Industrial and Applied Mathematics, 2019. URL: https://doi.org/10.1137/1.9781611975482.85.
  2. AmirMahdi Ahmadinejad, John Peebles, Edward Pyne, Aaron Sidford, and Salil Vadhan. Singular value approximation and sparsifying random walks on directed graphs. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 846-854, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00054.
  3. Zeyuan Allen-Zhu, Zhenyu Liao, and Lorenzo Orecchia. Spectral sparsification and regret minimization beyond matrix multiplicative updates. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 237-245, New York, NY, USA, 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2746539.2746610.
  4. Alexandr Andoni, Jiecao Chen, Robert Krauthgamer, Bo Qin, David P. Woodruff, and Qin Zhang. On sketching quadratic forms. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, ITCS '16, pages 311-319, New York, NY, USA, 2016. Association for Computing Machinery. URL: https://doi.org/10.1145/2840728.2840753.
  5. Afonso S Bandeira, March T Boedihardjo, and Ramon van Handel. Matrix concentration inequalities and free probability. Inventiones mathematicae, pages 1-69, 2023. Google Scholar
  6. Nikhil Bansal, Haotian Jiang, and Raghu Meka. Resolving matrix spencer conjecture up to poly-logarithmic rank. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 1814-1819, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585103.
  7. Joshua Batson, Daniel A Spielman, and Nikhil Srivastava. Twice-Ramanujan sparsifiers. SIAM Journal on Computing, 41(6):1704-1721, 2012. Google Scholar
  8. Joshua Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM Rev., 56(2):315-334, January 2014. URL: https://doi.org/10.1137/130949117.
  9. József Beck. Roth’s estimate of the discrepancy of integer sequences is nearly sharp. Combinatorica, 1(4):319-325, 1981. Google Scholar
  10. 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. Association for Computing Machinery. URL: https://doi.org/10.1145/237814.237827.
  11. Timothy Chu, Yu Gao, Richard Peng, Sushant Sachdeva, Saurabh Sawlani, and Junxing 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, 2018. URL: https://doi.org/10.1109/FOCS.2018.00042.
  12. Michael B. Cohen, Jonathan Kelner, Rasmus Kyng, John Peebles, Richard Peng, Anup B. Rao, and Aaron Sidford. Solving directed laplacian systems in nearly-linear time through sparse lu factorizations. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 898-909, 2018. URL: https://doi.org/10.1109/FOCS.2018.00089.
  13. Michael B. Cohen, Jonathan Kelner, John Peebles, Richard Peng, Anup B. Rao, Aaron Sidford, and Adrian Vladu. Almost-linear-time algorithms for markov chains and new spectral primitives for directed graphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 410-419, New York, NY, USA, 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3055399.3055463.
  14. Michael B. Cohen, Jonathan Kelner, John Peebles, Richard Peng, Aaron Sidford, and Adrian Vladu. Faster algorithms for computing the stationary distribution, simulating random walks, and more. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 583-592, 2016. URL: https://doi.org/10.1109/FOCS.2016.69.
  15. Daniel Dadush, Haotian Jiang, and Victor Reis. A new framework for matrix discrepancy: Partial coloring bounds via mirror descent. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 649-658, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3519935.3519967.
  16. David Durfee, Rasmus Kyng, John Peebles, Anup B Rao, and Sushant Sachdeva. Sampling random spanning trees faster than matrix multiplication. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 730-742. ACM, 2017. URL: https://arxiv.org/abs/1611.07451.
  17. 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. URL: https://arxiv.org/abs/1705.00985.
  18. Samuel B. Hopkins, Prasad Raghavendra, and Abhishek Shetty. Matrix discrepancy from quantum communication. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 637-648, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3519935.3519954.
  19. Arun Jambulapati, Victor Reis, and Kevin Tian. Linear-Sized Sparsifiers via Near-Linear Time Discrepancy Theory, pages 5169-5208. Society for Industrial and Applied Mathematics, 2023. URL: https://doi.org/10.1137/1.9781611977912.186.
  20. Arun Jambulapati and Aaron Sidford. Efficient Õ(n/ε) spectral sketches for the laplacian and its pseudoinverse. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '18, pages 2487-2503, USA, 2018. Society for Industrial and Applied Mathematics. Google Scholar
  21. Michael Kapralov and Rina Panigrahy. Spectral sparsification via random spanners. In Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, ITCS '12, pages 393-398, New York, NY, USA, 2012. Association for Computing Machinery. URL: https://doi.org/10.1145/2090236.2090267.
  22. Ioannis Koutis, Alex Levin, and Richard Peng. Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices. In Thomas Wilke Christoph Dürr, editor, STACS'12 (29th Symposium on Theoretical Aspects of Computer Science), volume 14, pages 266-277, Paris, France, February 2012. LIPIcs. URL: https://hal.science/hal-00678205.
  23. Ioannis Koutis and Shen Chen Xu. Simple parallel and distributed algorithms for spectral graph sparsification. ACM Trans. Parallel Comput., 3(2), August 2016. URL: https://doi.org/10.1145/2948062.
  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 Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC '16, pages 842-850, New York, NY, USA, 2016. Association for Computing Machinery. URL: https://doi.org/10.1145/2897518.2897640.
  25. Rasmus Kyng, Simon Meierhans, and Maximilian Probst Gutenberg. Derandomizing directed random walks in almost-linear time. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 407-418, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00046.
  26. Rasmus Kyng, Jakub Pachocki, Richard Peng, and Sushant Sachdeva. A Framework for Analyzing Resparsification Algorithms, pages 2032-2043. Society for Industrial and Applied Mathematics, 2017. URL: https://doi.org/10.1137/1.9781611974782.132.
  27. Rasmus Kyng and Sushant Sachdeva. Approximate gaussian elimination for laplacians - fast, sparse, and simple. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 573-582, October 2016. URL: https://doi.org/10.1109/FOCS.2016.68.
  28. Yin Tat Lee and He Sun. Constructing linear-sized spectral sparsification in almost-linear time. In Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), FOCS '15, pages 250-269, USA, 2015. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2015.24.
  29. Yin Tat Lee and He Sun. An sdp-based algorithm for linear-sized spectral sparsification. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 678-687, New York, NY, USA, 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3055399.3055477.
  30. Avi Levy, Harishchandra Ramadas, and Thomas Rothvoss. Deterministic discrepancy minimization via the multiplicative weight update method. In International Conference on Integer Programming and Combinatorial Optimization, pages 380-391. Springer, 2017. Google Scholar
  31. Huan Li and Zhongzhi Zhang. Kirchhoff index as a measure of edge centrality in weighted networks: Nearly linear time algorithms. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2377-2396. SIAM, 2018. URL: https://arxiv.org/abs/1708.05959.
  32. Yang P. Liu, Sushant Sachdeva, and Zejun Yu. Short cycles via low-diameter decompositions. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '19, pages 2602-2615, USA, 2019. Society for Industrial and Applied Mathematics. Google Scholar
  33. Adam W Marcus, Daniel A Spielman, and Nikhil Srivastava. Interlacing families ii: Mixed characteristic polynomials and the kadison—singer problem. Annals of Mathematics, pages 327-350, 2015. Google Scholar
  34. Raghu Meka. Discrepancy and beating the union bound. Windows on theory, a research blog, 2014. Google Scholar
  35. Merav Parter and Eylon Yogev. Optimal Short Cycle Decomposition in Almost Linear Time. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132 of Leibniz International Proceedings in Informatics (LIPIcs), pages 89:1-89:14, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.89.
  36. Richard Peng and Zhuoqing Song. Sparsified block elimination for directed laplacians. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 557-567, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3519935.3520053.
  37. Richard Peng and Daniel A. Spielman. An efficient parallel solver for sdd linear systems. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 333-342, New York, NY, USA, 2014. Association for Computing Machinery. URL: https://doi.org/10.1145/2591796.2591832.
  38. Victor Reis and Thomas Rothvoss. Linear size sparsifier and the geometry of the operator norm ball. In Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '20, pages 2337-2348, USA, 2020. Society for Industrial and Applied Mathematics. Google Scholar
  39. Victor Reis and Thomas Rothvoss. Vector balancing in lebesgue spaces. Random Structures & Algorithms, 62(3):667-688, 2023. Google Scholar
  40. Sushant Sachdeva and Yibin Zhao. A simple and efficient parallel laplacian solver. In Proceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '23, pages 315-325, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3558481.3591101.
  41. Thatchaphol Saranurak and Di Wang. Expander decomposition and pruning: Faster, stronger, and simpler. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '19, pages 2616-2635, USA, 2019. Society for Industrial and Applied Mathematics. Google Scholar
  42. 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.
  43. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC '08, pages 563-568, New York, NY, USA, 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1374376.1374456.
  44. Daniel A. Spielman and Shang-Hua Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, STOC '04, pages 81-90, New York, NY, USA, 2004. Association for Computing Machinery. URL: https://doi.org/10.1145/1007352.1007372.
  45. Daniel A. Spielman and Shang-Hua Teng. Spectral sparsification of graphs. SIAM J. Comput., 40(4):981-1025, July 2011. URL: https://doi.org/10.1137/08074489X.
  46. Joel A Tropp. User-friendly tail bounds for sums of random matrices. Foundations of computational mathematics, 12(4):389-434, 2012. Google Scholar
  47. Anastasios Zouzias. A matrix hyperbolic cosine algorithm and applications. In International Colloquium on Automata, Languages, and Programming, pages 846-858. Springer, 2012. Google Scholar
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