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Engineering a combinatorial Laplacian solver: lessons learned. (English) Zbl 07042381
Summary: Linear system solving is a main workhorse in applied mathematics. Recently, theoretical computer scientists contributed sophisticated algorithms for solving linear systems with symmetric diagonally-dominant (SDD) matrices in provably nearly-linear time. These algorithms are very interesting from a theoretical perspective, but their practical performance was unclear. Here, we address this gap. We provide the first implementation of the combinatorial solver by J. A. Kelner et al. [in: Proceedings of the 45th annual ACM symposium on theory of computing, STOC ’13. New York, NY: Association for Computing Machinery (ACM). 911–920 (2013; Zbl 1293.68145)], which is appealing for implementation due to its conceptual simplicity. The algorithm exploits that a Laplacian matrix (which is SDD) corresponds to a graph; solving symmetric Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles induced by a spanning tree with the low-stretch property. The results of our experiments are ambivalent. While they confirm the predicted nearly-linear running time, the constant factors make the solver much slower for reasonable inputs than basic methods with higher asymptotic complexity. We were also not able to use the solver effectively as a smoother or preconditioner. Moreover, while spanning trees with lower stretch indeed reduce the solver’s running time, we experience again a discrepancy in practice: in our experiments, simple spanning tree algorithms perform better than those with a guaranteed low stretch. We expect that our results provide insights for future improvements of combinatorial linear solvers.

MSC:
68W30 Symbolic computation and algebraic computation
05C21 Flows in graphs
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C85 Graph algorithms (graph-theoretic aspects)
65F05 Direct numerical methods for linear systems and matrix inversion
68R10 Graph theory (including graph drawing) in computer science
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