Telling Stories: Enumerating Maximal Directed Acyclic Graphs with Constrained Set of Sources and Targets

Abstract

In this chapter we propose an example of enumeration problem with a potentially exponential number of solutions that we solve by using a non-trivial brute force approach. The problem is a constrained version of the problem of enumerating all maximal directed acyclic subgraphs (DAG) of a graph $G$. In this version, we enumerate maximal DAGs whose sources and targets belong to a predefined subset of the vertices. We call such DAGs stories. Given a Metabolic network, each individual story should explain how some interesting metabolites are derived from some other through a chain of reactions, by keeping alternative pathways. We first show how to compute one story in polynomial-time, and then describe two different algorithms to “tell” all possible stories.