  This paper investigates the efficiency, robustness, and scalability of
approximate ideal restriction (AIR) algebraic multigrid as a preconditioner in
the all-at-once solution of a space-time hybridizable discontinuous Galerkin
(HDG) discretization of advection-dominated flows. The motivation for this
study is that the time-dependent advection-diffusion equation can be seen as a
"steady" advection-diffusion problem in $(d+1)$-dimensions and AIR has been
shown to be a robust solver for steady advection-dominated problems. Numerical
examples demonstrate the effectiveness of AIR as a preconditioner for
advection-diffusion problems on fixed and time-dependent domains, using both
slab-by-slab and all-at-once space-time discretizations, and in the context of
uniform and space-time adaptive mesh refinement. A closer look at the geometric
coarsening structure that arises in AIR also explains why AIR can provide
robust, scalable space-time convergence on advective and hyperbolic problems,
while most multilevel parallel-in-time schemes struggle with such problems.
