A fast algorithm to construct a representation for transversal matroids

Abstract

Transversal matroids were introduced in the mid 1960s and unified many results in transversal theory. Piff and Welsh proved that a transversal matroid is representable over all sufficiently large fields. To date, their merge algorithm is the only known algorithm to construct a representation matrix for a given transversal matroid. In this paper, a new algorithm to construct a representation matrix for a given transversal matroid is proposed that is faster than the Piff–Welsh algorithm. Let $$G=(V^{(1)}\dot{\cup }V^{(2)},E)$$ be the bipartite graph representing a transversal matroid and $${\mathcal {M}}$$ the set of all complete matchings of G. The time complexity of the proposed algorithm is $$O\left( |V^{(1)}|^{1/2}|E| + |V^{(1)}| |{\mathcal {M}}|\right) $$ . This algorithm makes use of complete matchings of bipartite graphs instead of matrix determinants, and an enumeration algorithm is used to find these matchings.