Hardness Results for Structured Linear Systems
This provides a theoretical barrier for algorithm designers working on fast linear system solvers.
The paper shows that extending nearly-linear time solvers for Laplacian matrices to slightly larger families would enable solving all real linear systems quickly, implying either broad progress or a halt in progress.
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.