Algebraic Multigrid Complexity for Polynomially Bounded SWCDDM Matrices

Prove that Algebraic Multigrid achieves near-linear complexity for polynomially bounded symmetric weakly-chained diagonally dominant M-matrices (SWCDDM): specifically, establish that for any polynomially bounded SWCDDM matrix M ∈ R^{n×n}, there exists an Algebraic Multigrid construction that runs in time \tilde{O}(nnz(M)) to produce a symmetric preconditioner Z that can be applied in O(nnz(M)) time and satisfies spectral equivalence Ω(1)·M^{-1} \preceq Z \preceq O(1)·M^{-1}, with constants independent of n.

Background

Multigrid methods achieve optimal O(n) behavior for classical elliptic PDE discretizations, but a general algebraic theory guaranteeing this performance beyond standard model problems is lacking. The authors define a broad algebraic class—polynomially bounded SWCDDM matrices—that resembles discretizations of elliptic PDEs and propose a theoretical analysis of Algebraic Multigrid on this class.

Establishing such a result would align goals in numerical analysis and theoretical computer science by providing provable, near-linear multigrid solvers for a large, practically relevant matrix class.

References

The conjecture we propose here is to prove the O(n) complexity of a multigrid method for a general class of linear systems with polynomially bounded SWCDDM matrices, described algebraically below.

— Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop  (2602.05394 - Amsel et al., 5 Feb 2026) in Subsection "Correctness of multigrid methods beyond the standard model problems" (Section 2)