Hardness Results for Structured Linear Systems

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.

Authors

• Rasmus Kyng
• Peng Zhang 0052

Keywords

• Linear systems
• Transmission line matrix methods
• Laplace equations
• Two dimensional displays
• Approximation algorithms
• Algorithm design and analysis
• Iterative methods
• Linear System
• Linear Equation
• Laplacian Matrix
• Running Time
• Fast System
• Accuracy Time
• Approximate Solution
• Sparse Matrix
• Conjugate Gradient
• Block Diagonal
• 2D Plane
• Generator Matrix
• Positive Semidefinite Matrix
• Interior Point Method
• Incidence Matrix
• Positive Semidefinite
• Class Of Matrices
• Matrix Integrity
• Normal Equations
• Vertex Coordinates
• Condition Number Of Matrix
• Auxiliary Equation
• Form Of Equation
• Total Variance
• Nonzero Entries
• Convex Combination
• Pairing
• New Variables
• Interior Point
• Numerical Linear Algebra
• Linear System Solvers
• Laplacian Solvers
• Multi-commodity Flow Problems
• Truss Stiffness Matrices
• Total Variation Matrices
• Complexity Theory
• Fine-grained Complexity