min_dominating_set.py

# -*- coding: utf-8 -*-
"""
**************************************
Minimum Vertex and Edge Dominating Set
**************************************


A dominating set for a graph G = (V, E) is a subset D of V such that every
vertex not in D is joined to at least one member of D by some edge. The
domination number gamma(G) is the number of vertices in a smallest dominating
set for G. Given a graph G = (V, E) find a minimum weight dominating set V'.

http://en.wikipedia.org/wiki/Dominating_set

An edge dominating set for a graph G = (V, E) is a subset D of E such that
every edge not in D is adjacent to at least one edge in D.

http://en.wikipedia.org/wiki/Edge_dominating_set
"""
#   Copyright (C) 2011-2012 by
#   Nicholas Mancuso <nick.mancuso@gmail.com>
#   All rights reserved.
#   BSD license.
import networkx as nx
__all__ = ["min_weighted_dominating_set",
           "min_edge_dominating_set"]
__author__ = """Nicholas Mancuso (nick.mancuso@gmail.com)"""


def min_weighted_dominating_set(G, weight=None):
    """Return minimum weight vertex dominating set.

    Parameters
    ----------
    G : NetworkX graph
      Undirected graph

    weight : None or string, optional (default = None)
        If None, every edge has p/distance/weight 1. If a string, use this
        edge attribute as the edge weight. Any edge attribute not present
        defaults to 1.

    Returns
    -------
    min_weight_dominating_set : set
      Returns a set of vertices whose weight sum is no more than log w(V) * OPT

    Notes
    -----
    This algorithm computes an approximate minimum weighted dominating set
    for the graph G. The upper-bound on the size of the solution is
    log w(V) * OPT.  Runtime of the algorithm is `O(|E|)`.

    References
    ----------
    .. [1] Vazirani, Vijay Approximation Algorithms (2001)
    """
    if not G:
        raise ValueError("Expected non-empty NetworkX graph!")

    # min cover = min dominating set
    weight = None
    dom_set = set([])
    cost_func = dict((node, nd.get(weight, 1)) \
                     for node, nd in G.nodes(data=True))
    

    vertices = set(G)
    sets = dict((node, set([node]) | set(G[node])) for node in G)

    def _cost(subset):
        """ Our cost effectiveness function for sets given its weight
        """
        cost = sum(cost_func[node] for node in subset)
        return cost / float(len(subset - dom_set))

    while vertices:
        # find the most cost effective set, and the vertex that for that set
        dom_node, min_set = min(sets.items(),
                                key=lambda x: _cost(x[1]))
        print("cost_func: ", cost_func)
        print("dom_node: ", dom_node)
        print("min_set: ", min_set)
        alpha = _cost(min_set)
        print("alpha: ", alpha)
        # reduce the cost for the rest
        for node in min_set - dom_set:
            cost_func[node] = alpha

        # add the node to the dominating set and reduce what we must cover
        dom_set.add(dom_node)
        del sets[dom_node]
        vertices = vertices - min_set

    return dom_set


def min_edge_dominating_set(G):
    """Return 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 nx.maximal_matching(G)