# Source code for networkx.algorithms.approximation.clique

```
"""Functions for computing large cliques and maximum independent sets."""
import networkx as nx
from networkx.algorithms.approximation import ramsey
from networkx.utils import not_implemented_for
__all__ = [
"clique_removal",
"max_clique",
"large_clique_size",
"maximum_independent_set",
]
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def maximum_independent_set(G):
"""Returns an approximate maximum independent set.
Independent set or stable set is a set of vertices in a graph, no two of
which are adjacent. That is, it is a set I of vertices such that for every
two vertices in I, there is no edge connecting the two. Equivalently, each
edge in the graph has at most one endpoint in I. The size of an independent
set is the number of vertices it contains [1]_.
A maximum independent set is a largest independent set for a given graph G
and its size is denoted $\\alpha(G)$. The problem of finding such a set is called
the maximum independent set problem and is an NP-hard optimization problem.
As such, it is unlikely that there exists an efficient algorithm for finding
a maximum independent set of a graph.
The Independent Set algorithm is based on [2]_.
Parameters
----------
G : NetworkX graph
Undirected graph
Returns
-------
iset : Set
The apx-maximum independent set
Examples
--------
>>> G = nx.path_graph(10)
>>> nx.approximation.maximum_independent_set(G)
{0, 2, 4, 6, 9}
Raises
------
NetworkXNotImplemented
If the graph is directed or is a multigraph.
Notes
-----
Finds the $O(|V|/(log|V|)^2)$ apx of independent set in the worst case.
References
----------
.. [1] `Wikipedia: Independent set
<https://en.wikipedia.org/wiki/Independent_set_(graph_theory)>`_
.. [2] Boppana, R., & Halldórsson, M. M. (1992).
Approximating maximum independent sets by excluding subgraphs.
BIT Numerical Mathematics, 32(2), 180–196. Springer.
"""
iset, _ = clique_removal(G)
return iset
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
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
Examples
--------
>>> G = nx.path_graph(10)
>>> nx.approximation.max_clique(G)
{8, 9}
Raises
------
NetworkXNotImplemented
If the graph is directed or is a multigraph.
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
"""
# 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]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def clique_removal(G):
r"""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).
Examples
--------
>>> G = nx.path_graph(10)
>>> nx.approximation.clique_removal(G)
({0, 2, 4, 6, 9}, [{0, 1}, {2, 3}, {4, 5}, {6, 7}, {8, 9}])
Raises
------
NetworkXNotImplemented
If the graph is directed or is a multigraph.
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")
@nx._dispatchable
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
-------
k: integer
The size of a large clique in the graph.
Examples
--------
>>> G = nx.path_graph(10)
>>> nx.approximation.large_clique_size(G)
2
Raises
------
NetworkXNotImplemented
If the graph is directed or is a multigraph.
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
```