  We introduce a new algorithm and software for solving linear equations in
symmetric diagonally dominant matrices with non-positive off-diagonal entries
(SDDM matrices), including Laplacian matrices. We use pre-conditioned conjugate
gradient (PCG) to solve the system of linear equations. Our preconditioner is a
variant of the Approximate Cholesky factorization of Kyng and Sachdeva (FOCS
2016). Our factorization approach is simple: we eliminate matrix rows/columns
one at a time and update the remaining matrix using sampling to approximate the
outcome of complete Cholesky factorization. Unlike earlier approaches, our
sampling always maintains a connectivity in the remaining non-zero structure.
Our algorithm comes with a tuning parameter that upper bounds the number of
samples made per original entry. We implement our algorithm in Julia, providing
two versions, AC and AC2, that respectively use 1 and 2 samples per original
entry. We compare their single-threaded performance to that of current
state-of-the-art solvers Combinatorial Multigrid (CMG),
BoomerAMG-preconditioned Krylov solvers from HyPre and PETSc, Lean Algebraic
Multigrid (LAMG), and MATLAB's with Incomplete Cholesky Factorization (ICC).
Our evaluation uses a broad class of problems, including all large SDDM
matrices from the SuiteSparse collection and diverse programmatically generated
instances. Our experiments suggest that our algorithm attains a level of
robustness and reliability not seen before in SDDM solvers, while retaining
good performance across all instances. Our code and data are public, and we
provide a tutorial on how to replicate our tests. We hope that others will
adopt this suite of tests as a benchmark, which we refer to as SDDM2023. Our
solver code is available at: https://github.com/danspielman/Laplacians.jl/ Our
benchmarking data and tutorial are available at:
https://rjkyng.github.io/SDDM2023/
