An improved algorithm for recognizing matroids

TL;DR

This paper introduces a new axiom system for matroids based on locked subsets, leading to an improved polynomial-time algorithm for recognizing matroids and uniform matroids, surpassing previous methods that relied on oracles.

Contribution

The paper presents a novel axiom system for matroids using locked subsets and develops a more efficient recognition algorithm, including a polynomial-time method for uniform matroids.

Findings

New axiom system for matroids based on locked subsets

Improved algorithm for recognizing matroids with better complexity

Polynomial-time recognition algorithm for uniform matroids

Abstract

Let $M$ be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. Locked subsets characterize nontrivial facets of the bases polytope. In this paper, we give a new axiom system for matroids based on locked subsets. We deduce an algorithm for recognizing matroids improving the running time complexity of the best known till today. This algorithm induces a polynomial time algorithm for recognizing uniform matroids. This latter problem is intractable if we use an independence oracle.