  Nisan showed in 1991 that the width of a smallest noncommutative
single-(source,sink) algebraic branching program (ABP) to compute a
noncommutative polynomial is given by the ranks of specific matrices. This
means that the set of noncommutative polynomials with ABP width complexity at
most $k$ is Zariski-closed, an important property in geometric complexity
theory. It follows that approximations cannot help to reduce the required ABP
width.
  It was mentioned by Forbes that this result would probably break when going
from single-(source,sink) ABPs to trace ABPs. We prove that this is correct.
Moreover, we study the commutative monotone setting and prove a result similar
to Nisan, but concerning the analytic closure. We observe the same behavior
here: The set of polynomials with ABP width complexity at most $k$ is closed
for single-(source,sink) ABPs and not closed for trace ABPs. The proofs reveal
an intriguing connection between tangent spaces and the vector space of flows
on the ABP. We close with additional observations on VQP and the closure of VNP
which allows us to establish a separation between the two classes.
