An Empirical Study of Cycle Toggling Based Laplacian Solvers
  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. 
Kevin Deweese, John R. Gilbert, Gary Miller, Richard Peng, Hao Ran Xu,   Shen Chen Xu
cs.DS