# A note on Cunningham's algorithm for matroid intersection

## TL;DR

This paper improves the implementation efficiency of Cunningham's algorithm for the matroid intersection problem, reducing the number of oracle calls needed to find the maximum common independent set.

## Contribution

It demonstrates that Cunningham's algorithm can be implemented with $O(nr ext{log}^2(r))$ independent oracle calls, enhancing its practical efficiency.

## Key findings

- Reduced oracle call complexity for Cunningham's algorithm
- Improved efficiency in solving the matroid intersection problem
- Potential for faster algorithms in related combinatorial optimization problems

## Abstract

In the matroid intersection problem, we are given two matroids of rank $r$ on a common ground set $E$ of $n$ elements and the goal is to find the maximum set that is independent in both matroids. In this note, we show that Cunningham's algorithm for matroid intersection can be implemented to use $O(nr\log^2(r))$ independent oracle calls.