Hardness Results for Structured Linear Systems

TL;DR

This paper explores the implications of extending nearly-linear time solvers for Laplacian matrices to broader classes of linear systems, highlighting potential breakthroughs or fundamental limitations in solving all linear systems efficiently.

Contribution

It establishes a conditional complexity result linking nearly-linear solvers for specific matrix families to the general problem of solving all linear systems.

Findings

Extending nearly-linear solvers to larger families implies solving all linear systems efficiently.

Progress in specialized solvers may be inherently limited by this complexity barrier.

The result frames a dichotomy in the development of fast linear system solvers.

Abstract

We show that if the nearly-linear time solvers for Laplacian matrices and their generalizations can be extended to solve just slightly larger families of linear systems, then they can be used to quickly solve all systems of linear equations over the reals. This result can be viewed either positively or negatively: either we will develop nearly-linear time algorithms for solving all systems of linear equations over the reals, or progress on the families we can solve in nearly-linear time will soon halt.