  An intersection digraph is a digraph where every vertex $v$ is represented by
an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$
if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive
if $S_v\cap T_v\neq \emptyset$ for every vertex $v$. Compared to well-known
undirected intersection graphs like interval graphs and permutation graphs, not
many algorithmic applications on intersection digraphs have been developed.
Motivated by the successful story on algorithmic applications of intersection
graphs using a graph width parameter called mim-width, we introduce its
directed analogue called `bi-mim-width' and prove that various classes of
reflexive intersection digraphs have bounded bi-mim-width. In particular, we
show that as a natural extension of $H$-graphs, reflexive $H$-digraphs have
linear bi-mim-width at most $12|E(H)|$, which extends a bound on the linear
mim-width of $H$-graphs [On the Tractability of Optimization Problems on
$H$-Graphs. Algorithmica 2020]. For applications, we introduce a novel
framework of directed versions of locally checkable problems, that streamlines
the definitions and the study of many problems in the literature and
facilitates their common algorithmic treatment. We obtain unified
polynomial-time algorithms for these problems on digraphs of bounded
bi-mim-width, when a branch decomposition is given. Locally checkable problems
include Kernel, Dominating Set, and Directed $H$-Homomorphism.
