Document Open Access Logo

A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems

Authors Monika Henzinger , Billy Jin , Richard Peng , David P. Williamson

Thumbnail PDF


  • Filesize: 1 MB
  • 22 pages

Document Identifiers

Author Details

Monika Henzinger
  • Faculty of Computer Science, Universität Wien, Austria
Billy Jin
  • School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, USA
Richard Peng
  • Cheriton School of Computer Science, University of Waterloo, Canada
David P. Williamson
  • School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, USA

Cite AsGet BibTex

Monika Henzinger, Billy Jin, Richard Peng, and David P. Williamson. A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 69:1-69:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form 𝐋𝐱 = 𝐛, where 𝐋 is the Laplacian matrix of a weighted graph with weights w(i,j) > 0 on the edges. The solution 𝐱 of the linear system can be interpreted as the potentials of an electrical flow in which the resistance on edge (i,j) is 1/w(i,j). Kelner, Orrechia, Sidford, and Zhu [Kelner et al., 2013] give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles (cycle toggling). In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by cut toggling: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a near-linear number of iterations. We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vector-matrix-vector problem (OMv), which has been conjectured to be difficult for dynamic algorithms [Henzinger et al., 2015]. The conjecture implies that the data structure does not have an O(n^{1-ε}) time algorithm for any ε > 0, and thus a straightforward implementation of the cut-toggling algorithm requires essentially linear time per iteration. To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an Õ(m^{1.5}) time cut-toggling algorithm for solving Laplacian systems. Furthermore, we show that if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, we can reduce the running time to O(m^{1 + ε}) for any ε > 0. Thus, the dual cut-toggling algorithm can achieve (almost) the same running time as its primal cycle-toggling counterpart.

Subject Classification

ACM Subject Classification
  • Theory of computation → Network flows
  • Laplacian solver
  • electrical flow
  • data structure


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


  1. Ittai Abraham and Ofer Neiman. Using petal-decompositions to build a low stretch spanning tree. Proceedings of the 44th Symposium on the Theory of Computing, pages 395-406, 2012. URL:
  2. Joshua D. Batson, Daniel A. Spielman, Nikhil Srivastava, and Shang-Hua Teng. Spectral sparsification of graphs: theory and algorithms. Commun. ACM, 56(8):87-94, 2013. Available at: URL:
  3. Erik G. Boman, Kevin Deweese, and John R. Gilbert. An empirical comparison of graph Laplacian solvers. In Proceedings of the 18th Workshop on Algorithm Engineering and Experiments, pages 174-188, 2016. Google Scholar
  4. Erik G. Boman, Kevin Deweese, and John R. Gilbert. Evaluating the potential of a dual randomized Kaczmarz Laplacian linear solver. Informatica, 40:95-107, 2016. Google Scholar
  5. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time, 2022. URL:
  6. Paul Christiano, Jonathan A. Kelner, Aleksander Mądry, Daniel Spielman, and Shang-Hua Teng. Electrical flows, Laplacian systems, and faster approximation of maximum flow in undirected graphs. In Proceedings of the 42nd Annual ACM Symposium on the Theory of Computing, pages 273-282, 2011. Google Scholar
  7. Kevin Deweese, John R. Gilbert, Gary Miller, Richard Peng, Hao Ran Xu, and Shen Chen Xu. An empirical study of cycle toggling based Laplacian solvers. In Proceedings of the 7th SIAM Workshop on Combinatorial Scientific Computing, pages 33-41, 2016. Google Scholar
  8. Thomas R. Ervolina and S. Thomas McCormick. Two strongly polynomial cut cancelling algorithms for minimum cost network flow. Discrete Applied Mathematics, 46:133-165, 1993. Google Scholar
  9. Yu Gao, Yang P. Liu, and Richard Peng. Fully dynamic electrical flows: Sparse maxflow faster than Goldberg-Rao. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 516-527, 2022. URL:
  10. Andrew V. Goldberg and Robert E. Tarjan. Finding minimum-cost circulations by canceling negative cycles. Journal of the ACM, 36:873-886, 1989. Google Scholar
  11. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. Proceedings of the 47th Annual ACM Symposium on the Theory of Computing, pages 21-30, 2015. URL:
  12. Daniel Hoske, Dimitar Lukarski, Henning Meyerhenke, and Michael Wegner. Engineering a combinatorial Laplacian solver: Lessons learned. Algorithms, 9, 2016. Article 72. Google Scholar
  13. David R. Karger. Minimum cuts in near-linear time. J. ACM, 47(1):46-76, January 2000. URL:
  14. Tarun Kathuria, Yang P. Liu, and Aaron Sidford. Unit capacity maxflow in almost m^4/3 time. SIAM Journal on Computing, 2022. To appear. URL:
  15. Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford, and Zeyuan Allen Zhu. A simple, combinatorial algorithm for solving SDD systems in nearly-linear time. Proceedings of the 45th Annual ACM Symposium on the Theory of Computing, pages 911-920, 2013. URL:
  16. Aleksander Mądry. Computing maximum flow with augmenting electrical flows. In Proceedings of the 57th IEEE Symposium on the Foundations of Computer Science, pages 593-602, 2016. Google Scholar
  17. Richard Peng and Daniel A. Spielman. An efficient parallel solver for SDD linear systems, 2013. URL:
  18. Jonah Sherman, October 2017. Personal communication. Google Scholar
  19. Daniel A. Spielman and Shang-Hua Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, pages 81-90, 2004. Google Scholar
  20. Daniel A. Spielman and Shang-Hua Teng. Spectral sparsification of graphs. SIAM J. Comput., 40(4):981-1025, 2011. Available at: URL:
  21. David P. Williamson. Network Flow Algorithms. Cambridge University Press, 2019. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail