Linear Representation of Transversal Matroids and Gammoids Parameterized by Rank

Abstract

Given a bipartite graph $G=(U\uplus V,E)$, a linear representation of the transversal matroid associated with G on the ground set U, can be constructed in randomized polynomial time. In fact one can get a linear representation deterministically in time $2^{{\mathcal {O}}(m^2n)}$, where $m=\vert U\vert$ and $n=|V|$, by looping through all the choices made in the randomized algorithm. Other important matroids for which one can obtain linear representation deterministically in time similar to the one for transversal matroids include gammoids and strict gammoids. Strict gammoids are duals of transversal matroids and gammoids are restrictions of strict gammoids. We give faster deterministic algorithms to construct linear representations of transversal matroids, gammoids and strict gammoids. All our algorithms run in time $\left( {\begin{array}{c}m\\ r\end{array}}\right) m^{{\mathcal {O}}(1)}$, where m is the cardinality of the ground set and r is the rank of the matroid. In the language of parameterized complexity, we give an $\mathsf{XP}$ algorithm for finding linear representations of transversal matroids, gammoids and strict gammoids parameterized by the rank of the given matroid.