Abstract
Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form ?? = ?, where ? is the Laplacian matrix of a weighted graph with weights w(i,j) > 0 on the edges. The solution ? 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, nearlinear 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 nearlinear 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 vectormatrixvector 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^{1ε}) time algorithm for any ε > 0, and thus a straightforward implementation of the cuttoggling 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 cuttoggling 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 cuttoggling algorithm can achieve (almost) the same running time as its primal cycletoggling counterpart.
BibTeX  Entry
@InProceedings{henzinger_et_al:LIPIcs.ITCS.2023.69,
author = {Henzinger, Monika and Jin, Billy and Peng, Richard and Williamson, David P.},
title = {{A Combinatorial CutToggling Algorithm for Solving Laplacian Linear Systems}},
booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
pages = {69:169:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772631},
ISSN = {18688969},
year = {2023},
volume = {251},
editor = {Tauman Kalai, Yael},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17572},
URN = {urn:nbn:de:0030drops175720},
doi = {10.4230/LIPIcs.ITCS.2023.69},
annote = {Keywords: Laplacian solver, electrical flow, data structure}
}
Keywords: 

Laplacian solver, electrical flow, data structure 
Collection: 

14th Innovations in Theoretical Computer Science Conference (ITCS 2023) 
Issue Date: 

2023 
Date of publication: 

01.02.2023 