Exact Matching in Matrix Multiplication Time

2025/08/06 by Sato, Ryotaro, Yamaguchi, Yutaro · 1 citation
#Combinatorics (math.CO) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Symbolic Computation (cs.SC)

paper · doi:10.48550/arxiv.2508.04081

Abstract

Initiated by Mulmuley, Vazirani, and Vazirani (1987), many algebraic algorithms have been developed for matching and related problems. In this paper, we review basic facts and discuss possible improvements with the aid of fast computation of the characteristic polynomial of a matrix. In particular, we show that the so-called exact matching problem can be solved with high probability in asymptotically the same time order as matrix multiplication. We also discuss its extension to the linear matroid parity problem.

Citations

• More Asymmetry Yields Faster Matrix Multiplication
• Determinantal Sieving
• Deterministic computation of the characteristic polynomial in the time of matrix multiplication
• A Weighted Linear Matroid Parity Algorithm
• Abusing the Tutte Matrix: An Algebraic Instance Compression for the K-set-cycle Problem
• La Budde's Method for Computing Characteristic Polynomials
• The Factorization of Linear Graphs
• Paths, Trees, and Flowers

Cited by

• Forgetting Alternation and Blossoms: A New Framework for Fast Matching Augmentation and Its Applications to Sequential/Distributed/Streaming Computation

Related

• Algebraic Structures and Algorithms for Matching and Matroid Problems (Preliminary Version)
• A Weighted Linear Matroid Parity Algorithm
• Deterministic computation of the characteristic polynomial in the time of matrix multiplication
• Fractional Linear Matroid Matching is in quasi-NC
• An Algorithmic Method of Partial Derivatives