An Algorithmic Proof of the Piff--Welsh Theorem on Transversal Matroid Representations

TL;DR

This paper transforms the classical, non-constructive proof of the Piff--Welsh theorem on transversal matroid representations into an explicit algorithmic procedure, enabling practical computation of such representations.

Contribution

It provides the first algorithmic version of the Piff--Welsh proof, making the theorem constructive and applicable in computational settings.

Findings

Algorithmic procedure for transversal matroid representation

Constructive proof applicable over any field characteristic

Facilitates computational applications of matroid theory

Abstract

A fundamental theorem of matroid theory establishes that a transversal matroid is representable over fields of any characteristic. It was proved in 1970 by Piff and Welsh: their proof is elegant and concise and, moveover, constructive. However it is far from being algorithmic, in terms of suggesting a step-by-step procedure for deriving a collection of vectors over a given base field representing the transversal matroid induced by a given set system. In this note we recast Piff and Welsh's proof in algorithmic form.