  We call a pair of closed subgroups $(G_1,G_2)$ from a connected reductive
algebraic group $G$ a {\it complexity $c$ pair} if the multiplication action of
the pair on $G$ is of complexity $c$. The main focus of this article is on the
cases where $G$ is simple and $c$ is either 0 or 1. After showing that both of
the subgroups $G_1$ and $G_2$ cannot be reductive subgroups unless $c>1$, we
look for the cases where exactly one of the subgroups $G_1$ and $G_2$ is
reductive. It turns out that there are only a few such pairs, and their
classification involves the horospherical homogeneous spaces of small ranks. As
a byproduct of the circle of ideas that we use for this development, we obtain
the classification of the diagonal spherical actions of simple algebraic groups
on the products of flag varieties with affine homogeneous spaces.
