Is Nearly-linear the same in Theory and Practice? A Case Study with a Combinatorial Laplacian Solver

TL;DR

This paper implements and evaluates a nearly-linear time combinatorial Laplacian solver, revealing practical performance gaps despite theoretical efficiency, and highlighting the impact of spanning tree choices on solver speed.

Contribution

First implementation and experimental evaluation of a combinatorial Laplacian solver based on low-stretch spanning trees, bridging theory and practice.

Findings

Nearly-linear time confirmed in practice, but with high constant factors.

Lower stretch spanning trees reduce solver runtime.

Simple spanning tree algorithms outperform guaranteed low-stretch methods.

Abstract

Linear system solving is one of the main workhorses in applied mathematics. Recently, theoretical computer scientists have contributed sophisticated algorithms for solving linear systems with symmetric diagonally dominant matrices (a class to which Laplacian matrices belong) in provably nearly-linear time. While these algorithms are highly interesting from a theoretical perspective, there are no published results how they perform in practice. With this paper we address this gap. We provide the first implementation of the combinatorial solver by [Kelner et al., STOC 2013], which is particularly appealing for implementation due to its conceptual simplicity. The algorithm exploits that a Laplacian matrix corresponds to a graph; solving 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 comprehensive experimental study are ambivalent. They confirm a nearly-linear running time, but for reasonable inputs the constant factors make the solver much slower than methods with higher asymptotic complexity. One other aspect predicted by theory is confirmed by our findings, though: Spanning trees with lower stretch indeed reduce the solver's running time. Yet, simple spanning tree algorithms perform in practice better than those with a guaranteed low stretch.