Exact Matching in Matrix Multiplication Time

TL;DR

This paper demonstrates that the exact matching problem can be solved with high probability in asymptotic time comparable to matrix multiplication, leveraging algebraic algorithms and characteristic polynomial computations.

Contribution

It shows that exact matching can be achieved efficiently using algebraic methods, extending to the linear matroid parity problem, with potential improvements in algorithmic complexity.

Findings

Exact matching solvable in near matrix multiplication time

High probability algorithms for algebraic matching problems

Extension to linear matroid parity problem

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.