An Empirical Study of Cycle Toggling Based Laplacian Solvers

TL;DR

This paper empirically evaluates cycle toggling based Laplacian solvers, demonstrating significant speedups and competitiveness with standard methods for specific graph structures.

Contribution

It provides a detailed empirical analysis of cycle toggling methods, introducing optimized algorithms for fundamental cycle adjustments in Laplacian solvers.

Findings

Significant speedups over previous implementations.

Competitive performance with standard numerical routines.

Effective handling of fundamental cycle adjustments.

Abstract

We study the performance of linear solvers for graph Laplacians based on the combinatorial cycle adjustment methodology proposed by [Kelner-Orecchia-Sidford-Zhu STOC-13]. The approach finds a dual flow solution to this linear system through a sequence of flow adjustments along cycles. We study both data structure oriented and recursive methods for handling these adjustments. The primary difficulty faced by this approach, updating and querying long cycles, motivated us to study an important special case: instances where all cycles are formed by fundamental cycles on a length $n$ path. Our methods demonstrate significant speedups over previous implementations, and are competitive with standard numerical routines.