Deterministic construction of exact linear representations for gammoids

Construct exact linear representations of gammoids in deterministic polynomial time, given a directed graph, a source set, and the gammoid’s ground set.

Background

The randomized cut-covering framework relies on linear representations of gammoids, whose known polynomial-time construction uses polynomial identity testing. A deterministic polynomial-time construction of such exact representations would directly derandomize the cut-covering lemma and the resulting kernelization framework.

The paper does not resolve this problem. Instead, it circumvents the need for exact representations by constructing deterministic almost multilinear representations of gammoids, which approximate matroid ranks and suffice for computing representative families of the required size. Thus, the existence of a deterministic polynomial-time construction for exact linear gammoid representations remains unresolved.