  This paper focuses on developing a reduction-based algebraic multigrid method
that is suitable for solving general (non)symmetric linear systems and is
naturally robust from pure advection to pure diffusion. Initial motivation
comes from a new reduction-based algebraic multigrid (AMG) approach, $\ell$AIR
(local approximate ideal restriction), that was developed for solving
advection-dominated problems. Though this new solver is very effective in the
advection dominated regime, its performance degrades in cases where diffusion
becomes dominant. This is consistent with the fact that in general,
reduction-based AMG methods tend to suffer from growth in complexity and/or
convergence rates as the problem size is increased, especially for diffusion
dominated problems in two or three dimensions. Motivated by the success of
$\ell$AIR in the advective regime, our aim in this paper is to generalize the
AIR framework with the goal of improving the performance of the solver in
diffusion dominated regimes. To do so, we propose a novel way to combine mode
constraints as used commonly in energy minimization AMG methods with the local
approximation of ideal operators used in $\ell$AIR. The resulting constrained
$\ell$AIR (C$\ell$AIR) algorithm is able to achieve fast scalable convergence
on advective and diffusive problems. In addition, it is able to achieve
standard low complexity hierarchies in the diffusive regime through aggressive
coarsening, something that has been previously difficult for reduction-based
methods.
