eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-02-01
69:1
69:22
10.4230/LIPIcs.ITCS.2023.69
article
A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems
Henzinger, Monika
1
https://orcid.org/0000-0002-5008-6530
Jin, Billy
2
https://orcid.org/0000-0002-6362-2048
Peng, Richard
3
https://orcid.org/0000-0002-5407-7965
Williamson, David P.
2
https://orcid.org/0000-0002-2884-0058
Faculty of Computer Science, Universität Wien, Austria
School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, USA
Cheriton School of Computer Science, University of Waterloo, Canada
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol251-itcs2023/LIPIcs.ITCS.2023.69/LIPIcs.ITCS.2023.69.pdf
Laplacian solver
electrical flow
data structure