# -*- coding: utf-8 -*-
# Copyright (C) 2011-2018 by
# Nicholas Mancuso <nick.mancuso@gmail.com>
# All rights reserved.
# BSD license.
# Copyright 2016-2018 NetworkX developers.
# NetworkX is distributed under a BSD license
#
# Authors: Nicholas Mancuso (nick.mancuso@gmail.com)
# Jeffery Finkelstein <jeffrey.finkelstein@gmail.com>
# Dan Schult <dschult@colgate.edu>
"""Functions for computing large cliques."""
from operator import itemgetter
import networkx as nx
from networkx.utils import not_implemented_for
from networkx.algorithms.approximation import ramsey
__all__ = ["clique_removal", "max_clique", "large_clique_size"]
[docs]def max_clique(G):
r"""Find the Maximum Clique
Finds the $O(|V|/(log|V|)^2)$ apx of maximum clique/independent set
in the worst case.
Parameters
----------
G : NetworkX graph
Undirected graph
Returns
-------
clique : set
The apx-maximum clique of the graph
Notes
------
A clique in an undirected graph G = (V, E) is a subset of the vertex set
`C \subseteq V` such that for every two vertices in C there exists an edge
connecting the two. This is equivalent to saying that the subgraph
induced by C is complete (in some cases, the term clique may also refer
to the subgraph).
A maximum clique is a clique of the largest possible size in a given graph.
The clique number `\omega(G)` of a graph G is the number of
vertices in a maximum clique in G. The intersection number of
G is the smallest number of cliques that together cover all edges of G.
https://en.wikipedia.org/wiki/Maximum_clique
References
----------
.. [1] Boppana, R., & Halldórsson, M. M. (1992).
Approximating maximum independent sets by excluding subgraphs.
BIT Numerical Mathematics, 32(2), 180–196. Springer.
doi:10.1007/BF01994876
"""
if G is None:
raise ValueError("Expected NetworkX graph!")
# finding the maximum clique in a graph is equivalent to finding
# the independent set in the complementary graph
cgraph = nx.complement(G)
iset, _ = clique_removal(cgraph)
return iset
[docs]def clique_removal(G):
""" Repeatedly remove cliques from the graph.
Results in a $O(|V|/(\log |V|)^2)$ approximation of maximum clique
and independent set. Returns the largest independent set found, along
with found maximal cliques.
Parameters
----------
G : NetworkX graph
Undirected graph
Returns
-------
max_ind_cliques : (set, list) tuple
2-tuple of Maximal Independent Set and list of maximal cliques (sets).
References
----------
.. [1] Boppana, R., & Halldórsson, M. M. (1992).
Approximating maximum independent sets by excluding subgraphs.
BIT Numerical Mathematics, 32(2), 180–196. Springer.
"""
graph = G.copy()
c_i, i_i = ramsey.ramsey_R2(graph)
cliques = [c_i]
isets = [i_i]
while graph:
graph.remove_nodes_from(c_i)
c_i, i_i = ramsey.ramsey_R2(graph)
if c_i:
cliques.append(c_i)
if i_i:
isets.append(i_i)
# Determine the largest independent set as measured by cardinality.
maxiset = max(isets, key=len)
return maxiset, cliques
[docs]@not_implemented_for('directed')
@not_implemented_for('multigraph')
def large_clique_size(G):
"""Find the size of a large clique in a graph.
A *clique* is a subset of nodes in which each pair of nodes is
adjacent. This function is a heuristic for finding the size of a
large clique in the graph.
Parameters
----------
G : NetworkX graph
Returns
-------
int
The size of a large clique in the graph.
Notes
-----
This implementation is from [1]_. Its worst case time complexity is
:math:`O(n d^2)`, where *n* is the number of nodes in the graph and
*d* is the maximum degree.
This function is a heuristic, which means it may work well in
practice, but there is no rigorous mathematical guarantee on the
ratio between the returned number and the actual largest clique size
in the graph.
References
----------
.. [1] Pattabiraman, Bharath, et al.
"Fast Algorithms for the Maximum Clique Problem on Massive Graphs
with Applications to Overlapping Community Detection."
*Internet Mathematics* 11.4-5 (2015): 421--448.
<https://doi.org/10.1080/15427951.2014.986778>
See also
--------
:func:`networkx.algorithms.approximation.clique.max_clique`
A function that returns an approximate maximum clique with a
guarantee on the approximation ratio.
:mod:`networkx.algorithms.clique`
Functions for finding the exact maximum clique in a graph.
"""
degrees = G.degree
def _clique_heuristic(G, U, size, best_size):
if not U:
return max(best_size, size)
u = max(U, key=degrees)
U.remove(u)
N_prime = {v for v in G[u] if degrees[v] >= best_size}
return _clique_heuristic(G, U & N_prime, size + 1, best_size)
best_size = 0
nodes = (u for u in G if degrees[u] >= best_size)
for u in nodes:
neighbors = {v for v in G[u] if degrees[v] >= best_size}
best_size = _clique_heuristic(G, neighbors, 1, best_size)
return best_size