Is Nearly-linear the Same in Theory and Practice? A Case Study with a Combinatorial Laplacian Solver

Abstract

Abstract. Linear system solving is one of the main workhorses in applied mathematics. Recently, theoretical computer scientists have contributed sophisticated algorithms for solving linear systems with symmetric diagonally dominant matrices (a class to which Laplacian matrices belong) in provably nearly-linear time. While these algorithms are highly interesting from a theoretical perspective, there are no published results how they perform in practice. With this paper we address this gap. We provide the first implementation of the combinatorial solver by [Kelner et al., STOC 2013], which is particularly appealing for implementation due to its conceptual simplicity. The algorithm exploits that a Laplacian matrix corresponds to a graph; solving Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles induced by a spanning tree with the low-stretch property. The results of our comprehensive experimental study are ambivalent. They confirm a nearlylinear running time, but for reasonable inputs the constant factors make the solver much slower than methods with higher asymptotic complexity. One other aspect predicted by theory is confirmed by our findings, though: Spanning trees with lower stretch indeed reduce the solver's running time. Yet, simple spanning tree algorithms perform in practice better than those with a guaranteed low stretch.

Evolving Difficult Graphs for Laplacian Solvers

“…KOSZ outperformed Jacobi PCG on several of these problems, which was an interesting discovery as previous experiments indicated pessimism regarding KOSZ performance [1,10]. With this knowledge we asked on what other problems might KOSZ outperform traditional methods and by how much.…”

“…KOSZ [16] was one of the first asymptotically fast Laplacian solvers to actually be implemented [1,10] to its relative simplicity. We skip many details here, but the algorithm randomly selects cycles from the graph and updates information on the edges of these cycles.…”

“…When solvers are introduced, sometimes they are accompanied by experimental results [6,15]. Other times additional experimental studies are conducted to evaluate the performance of existing or proposed solvers [2,10]. In either case, most experimental work in this area involves the application of solvers to test problems taken from specific research domains, large graph repositories, or sets of randomly generated graphs, with the solvers being the primary focus.…”

Engineering a Combinatorial Laplacian Solver: Lessons Learned

“…In this paper, which extends the previous conference version [30], we implement the KOSZ solver (the acronym follows from the authors' last names) by Kelner et al [12] and investigate its practical performance. To this end, we start in Section 2 by describing important notation and outlining KOSZ.…”

Efficient and Practical Tree Preconditioning for Solving Laplacian Systems

“…In the past several years, this basic framework for solving linear systems with Laplacian matrices has been extended significantly, with two major research directions: finding trees that can optimize the stretch with respect to arbitrary large graphs [1], and changing this basic framework to use a more sophisticated hierarchical graph approximation scheme in which preconditioners themselves can be solved via iterative (and possibly recursive) schemes [16]. Unfortunately, both of these directions lead to highly complex algorithms and their practical performance has been evaluated only very recently [13]. Rather than trying to improve either of these two directions, our goal is to show that a simple modification to the tree preconditioner can significantly improve the performance of the iterative solver both in theory (for some restricted cases) and in practice (over a large number of graph classes).…”