Dynamic Rank, Basis, and Matching

2026-05-10

AI Summary

This work addresses the problem of efficiently maintaining the rank, a column basis, and a maximum-rank submatrix of a matrix under dynamic updates—either to individual entries or entire columns—and applies these techniques to the dynamic maximum matching problem in graphs. The paper presents the first dynamic algorithm whose update time depends on the current rank \( r \) rather than the matrix dimension \( n \). By integrating sparse update strategies with rank-sensitive complexity analysis, it achieves an amortized update time of \( \tilde{O}(r^{1.405}) \) for single-entry modifications and \( \tilde{O}(r^{1.528} + z) \) for column updates, where \( z \) denotes the number of changed entries. This approach is the first to simultaneously support dynamic maintenance of rank, basis, and maximum-rank submatrix, yielding an edge update time of \( \tilde{O}(|M|^{1.405}) \) for dynamic graph matching and significantly improving upon prior methods.

Abstract

We study dynamic algorithms for maintaining fundamental algebraic properties of matrices, specifically, rank, basis, and full-rank submatrices, with applications to maximum matching on dynamic graphs. Prior dynamic algorithms for rank achieve subquadratic update times but scale with the matrix dimension $n$, and could not always maintain the corresponding objects such as a basis or maximum full-rank submatrix. We present the first dynamic rank algorithms whose update time scales with the matrix rank $r$, achieving $\tilde O(r^{1.405})$ time per entry-update and $\tilde O(r^{1.528}+ z)$ per column-update, where $z$ is the number of changed entries. This extends to $\tilde O(|M|^{1.405})$ edge-update time to maintain the size $|M|$ of a maximum matching. We also give dynamic algorithms for maintaining a column-basis subject to column-updates and a maximum full-rank submatrix subject to entry-updates.

Problem

• dynamic rank
• matrix basis
• full-rank submatrix
• maximum matching
• dynamic graphs

Innovation

• dynamic rank
• matrix basis
• maximum matching
• subquadratic update
• full-rank submatrix