Algorithms for generating convex sets in acyclic digraphs
P. Balister, S. Gerke, Gregory Gutin, A. Johnstone, J. Reddington, E. Scott, A. Soleimanfallah and A. Yeo
A set $X$ of vertices of an acyclic digraph $D$ is convex if $X\neq
\emptyset$ and there is no directed path between vertices of $X$
which contains a vertex not in $X$. A set $X$ is connected if $X\neq
\emptyset$ and the underlying undirected graph of the subgraph of
$D$ induced by $X$ is connected. Connected convex sets and convex
sets of acyclic digraphs are of interest in the area of modern
embedded processor technology. We construct an algorithm $\cal A$
for enumeration of all connected convex sets of an acyclic digraph
$D$ of order $n$. The time complexity of $\cal A$ is $O(n\cdot
cc(D))$, where $cc(D)$ is the number of connected convex sets in
$D$. We also give an optimal algorithm for enumeration of all (not
just connected) convex sets of an acyclic digraph $D$ of order $n$.
In computational experiments we demonstrate that our algorithms
outperform the best algorithms in the literature.