In this work we plan to revise the main techniques for enumeration
algorithms and to show four examples of enumeration algorithms that can
be applied to efficiently deal with some biological problems modelled by
using biological networks: enumerating central and peripheral nodes of a
network, enumerating stories, enumerating paths or cycles, and
enumerating bubbles. Notice that the corresponding computational
problems we define are of more general interest and our results hold in
the case of arbitrary graphs. Enumerating all the most and less central
vertices in a network according to their eccentricity is an example of
an enumeration problem whose solutions are polynomial and can be listed
in polynomial time, very often in linear or almost linear time in
practice. Enumerating stories, i.e. all maximal directed acyclic
subgraphs of a graph G whose sources and targets belong to a predefined
subset of the vertices, is on the other hand an example of an
enumeration problem with an exponential number of solutions, that can be
solved by using a non trivial brute-force approach. Given a metabolic
network, each individual story should explain how some interesting
metabolites are derived from some others through a chain of reactions,
by keeping all alternative pathways between sources and targets.
Enumerating cycles or paths in an undirected graph, such as a
protein-protein interaction undirected network, is an example of an
enumeration problem in which all the solutions can be listed through an
optimal algorithm, i.e. the time required to list all the solutions is
dominated by the time to read the graph plus the time required to print
all of them. By extending this result to directed graphs, it would be
possible to deal more efficiently with feedback loops and signed paths
analysis in signed or interaction directed graphs, such as gene
regulatory networks. Finally, enumerating mouths or bubbles with a
source s in a directed graph, that is enumerating all the two
vertex-disjoint directed paths between the source s and all the possible
targets, is an example of an enumeration problem in which all the
solutions can be listed through a linear delay algorithm, meaning that
the delay between any two consecutive solutions is linear, by turning
the problem into a constrained cycle enumeration problem. Such patterns,
in a de Bruijn graph representation of the reads obtained by sequencing,
are related to polymorphisms in DNA- or RNA-seq data.
