"""Functions for finding node and edge dominating sets.
A `dominating set`_ for an undirected graph *G* with vertex set *V*
and edge set *E* is a subset *D* of *V* such that every vertex not in
*D* is adjacent to at least one member of *D*. An `edge dominating set`_
is a subset *F* of *E* such that every edge not in *F* is
incident to an endpoint of at least one edge in *F*.
.. _dominating set: https://en.wikipedia.org/wiki/Dominating_set
.. _edge dominating set: https://en.wikipedia.org/wiki/Edge_dominating_set
"""
from ..matching import maximal_matching
from ...utils import not_implemented_for
__all__ = ["min_weighted_dominating_set", "min_edge_dominating_set"]
# TODO Why doesn't this algorithm work for directed graphs?
[docs]@not_implemented_for("directed")
def min_weighted_dominating_set(G, weight=None):
r"""Returns a dominating set that approximates the minimum weight node
dominating set.
Parameters
----------
G : NetworkX graph
Undirected graph.
weight : string
The node attribute storing the weight of an node. If provided,
the node attribute with this key must be a number for each
node. If not provided, each node is assumed to have weight one.
Returns
-------
min_weight_dominating_set : set
A set of nodes, the sum of whose weights is no more than `(\log
w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of
each node in the graph and `w(V^*)` denotes the sum of the
weights of each node in the minimum weight dominating set.
Notes
-----
This algorithm computes an approximate minimum weighted dominating
set for the graph `G`. The returned solution has weight `(\log
w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of each
node in the graph and `w(V^*)` denotes the sum of the weights of
each node in the minimum weight dominating set for the graph.
This implementation of the algorithm runs in $O(m)$ time, where $m$
is the number of edges in the graph.
References
----------
.. [1] Vazirani, Vijay V.
*Approximation Algorithms*.
Springer Science & Business Media, 2001.
"""
# The unique dominating set for the null graph is the empty set.
if len(G) == 0:
return set()
# This is the dominating set that will eventually be returned.
dom_set = set()
def _cost(node_and_neighborhood):
"""Returns the cost-effectiveness of greedily choosing the given
node.
`node_and_neighborhood` is a two-tuple comprising a node and its
closed neighborhood.
"""
v, neighborhood = node_and_neighborhood
return G.nodes[v].get(weight, 1) / len(neighborhood - dom_set)
# This is a set of all vertices not already covered by the
# dominating set.
vertices = set(G)
# This is a dictionary mapping each node to the closed neighborhood
# of that node.
neighborhoods = {v: {v} | set(G[v]) for v in G}
# Continue until all vertices are adjacent to some node in the
# dominating set.
while vertices:
# Find the most cost-effective node to add, along with its
# closed neighborhood.
dom_node, min_set = min(neighborhoods.items(), key=_cost)
# Add the node to the dominating set and reduce the remaining
# set of nodes to cover.
dom_set.add(dom_node)
del neighborhoods[dom_node]
vertices -= min_set
return dom_set
[docs]def min_edge_dominating_set(G):
r"""Returns minimum cardinality edge dominating set.
Parameters
----------
G : NetworkX graph
Undirected graph
Returns
-------
min_edge_dominating_set : set
Returns a set of dominating edges whose size is no more than 2 * OPT.
Notes
-----
The algorithm computes an approximate solution to the edge dominating set
problem. The result is no more than 2 * OPT in terms of size of the set.
Runtime of the algorithm is $O(|E|)$.
"""
if not G:
raise ValueError("Expected non-empty NetworkX graph!")
return maximal_matching(G)