Solving Graph Laplacian Systems Through Recursive Partitioning and Two-Grid Preconditioning

We present a parallelizable direct method for computing the solution to graph Laplacian-based linear systems derived from graphs that can be hierarchically bipartitioned with small edge cuts. For a graph of size $n$ that can be recursively partitioned into equal-sized sets with constant-size edge cuts, our method decomposes a graph Laplacian in time $O(n \log n)$ and then uses that decomposition to perform a linear solve in time $O(n \log n)$. Many real-world graphs, however, do not possess this property. So, we use this technique to design a preconditioner for graph Laplacians that do not have this property. Finally, we augment this preconditioner with a two-grid method that accounts for much of the preconditioner's weaknesses. We present an analysis of this method, as well as a general theorem for the condition number of a general class of two-grid support graph-based preconditioners. Numerical experiments illustrate the performance of the studied methods.