A combinatorial cut-toggling algorithm for solving Laplacian linear systems

Henzinger MH, Jin B, Peng R, Williamson DP. 2023. A combinatorial cut-toggling algorithm for solving Laplacian linear systems. Algorithmica. 85, 2680–3716.


Journal Article | Published | English

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Author
Henzinger, MonikaISTA ; Jin, Billy; Peng, Richard; Williamson, David P.
Abstract
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 et al. (in: Proceedings of the 45th Annual ACM Symposium on the Theory of Computing, pp 911–920, 2013. https://doi.org/10.1145/2488608.2488724) 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, which has been conjectured to be difficult for dynamic algorithms (Henzinger et al., in: Proceedings of the 47th Annual ACM Symposium on the Theory of Computing, pp 21–30, 2015. https://doi.org/10.1145/2746539.2746609). The conjecture implies that the data structure does not have an O(n1−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 O˜(m1.5) 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(m1+ϵ) for any ϵ>0. Thus, the dual cut-toggling algorithm can achieve (almost) the same running time as its primal cycle-toggling counterpart.
Publishing Year
Date Published
2023-12-01
Journal Title
Algorithmica
Acknowledgement
Monika Henzinger was supported by funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme Grant agreement No. 101019564 “The Design of Modern Fully Dynamic Data Structures (MoDynStruct)” and from the Austrian Science Fund (FWF) project “Fast Algorithms for a Reactive Network Layer (ReactNet)”, P 33775-N, with additional funding from the netidee SCIENCE Stiftung, 2020–2024. Billy Jin was Supported in part by NSERC fellowship PGSD3-532673-2019 and NSF grant CCF-2007009. Richard Peng was supported in part by an NSERC Discovery Grant and NSF grant CCF-1846218. David P. Williamson was supported in part by NSF grant CCF-2007009.
Volume
85
Page
2680-3716
ISSN
eISSN
IST-REx-ID

Cite this

Henzinger MH, Jin B, Peng R, Williamson DP. A combinatorial cut-toggling algorithm for solving Laplacian linear systems. Algorithmica. 2023;85:2680-3716. doi:10.1007/s00453-023-01154-8
Henzinger, M. H., Jin, B., Peng, R., & Williamson, D. P. (2023). A combinatorial cut-toggling algorithm for solving Laplacian linear systems. Algorithmica. Springer Nature. https://doi.org/10.1007/s00453-023-01154-8
Henzinger, Monika H, Billy Jin, Richard Peng, and David P. Williamson. “A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems.” Algorithmica. Springer Nature, 2023. https://doi.org/10.1007/s00453-023-01154-8.
M. H. Henzinger, B. Jin, R. Peng, and D. P. Williamson, “A combinatorial cut-toggling algorithm for solving Laplacian linear systems,” Algorithmica, vol. 85. Springer Nature, pp. 2680–3716, 2023.
Henzinger MH, Jin B, Peng R, Williamson DP. 2023. A combinatorial cut-toggling algorithm for solving Laplacian linear systems. Algorithmica. 85, 2680–3716.
Henzinger, Monika H., et al. “A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems.” Algorithmica, vol. 85, Springer Nature, 2023, pp. 2680–3716, doi:10.1007/s00453-023-01154-8.
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