  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.
