# Copyright 2018 D-Wave Systems Inc.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
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# distributed under the License is distributed on an "AS IS" BASIS,
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# See the License for the specific language governing permissions and
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# ================================================================================================
from __future__ import division
import networkx as nx
import dwave_networkx as dnx
from dwave_networkx.utils import binary_quadratic_model_sampler
__all__ = ["maximum_clique", "clique_number", "is_clique"]
[docs]@binary_quadratic_model_sampler(1)
def maximum_clique(G, sampler=None, lagrange=2.0, **sampler_args):
"""
Returns an approximate maximum clique.
A clique in an undirected graph, G = (V, E), is a subset of the vertex set
:math:`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.
This function works by finding the maximum independent set of the compliment
graph of the given graph G which is equivalent to finding maximum clique.
It defines a QUBO with ground states corresponding
to a maximum weighted independent set and uses the sampler to sample from it.
Parameters
----------
G : NetworkX graph
The graph on which to find a maximum clique.
sampler
A binary quadratic model sampler. A sampler is a process that
samples from low energy states in models defined by an Ising
equation or a Quadratic Unconstrained Binary Optimization
Problem (QUBO). A sampler is expected to have a 'sample_qubo'
and 'sample_ising' method. A sampler is expected to return an
iterable of samples, in order of increasing energy. If no
sampler is provided, one must be provided using the
`set_default_sampler` function.
lagrange : optional (default 2)
Lagrange parameter to weight constraints (no edges within set)
versus objective (largest set possible).
sampler_args
Additional keyword parameters are passed to the sampler.
Returns
-------
clique_nodes : list
List of nodes that form a maximum clique, as
determined by the given sampler.
Notes
-----
Samplers by their nature may not return the optimal solution. This
function does not attempt to confirm the quality of the returned
sample.
References
----------
`Maximum Clique on Wikipedia <https://en.wikipedia.org/wiki/Maximum_clique(graph_theory)>`_
`Independent Set on Wikipedia <https://en.wikipedia.org/wiki/Independent_set_(graph_theory)>`_
`QUBO on Wikipedia <https://en.wikipedia.org/wiki/Quadratic_unconstrained_binary_optimization>`_
.. [AL] Lucas, A. (2014). Ising formulations of many NP problems.
Frontiers in Physics, Volume 2, Article 5.
"""
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
complement_G = nx.complement(G)
return dnx.maximum_independent_set(complement_G, sampler, lagrange, **sampler_args)
[docs]@binary_quadratic_model_sampler(1)
def clique_number(G, sampler=None, lagrange=2.0, **sampler_args):
"""
Returns the number of vertices in the maximum clique of a graph.
A maximum clique is a clique of the largest possible size in a given graph.
The clique number math:`\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.
This function works by finding the maximum independent set of the compliment
graph of the given graph G which is equivalent to finding maximum clique.
It defines a QUBO with ground states corresponding
to a maximum weighted independent set and uses the sampler to sample from it.
Parameters
----------
G : NetworkX graph
The graph on which to find a maximum clique.
sampler
A binary quadratic model sampler. A sampler is a process that
samples from low energy states in models defined by an Ising
equation or a Quadratic Unconstrained Binary Optimization
Problem (QUBO). A sampler is expected to have a 'sample_qubo'
and 'sample_ising' method. A sampler is expected to return an
iterable of samples, in order of increasing energy. If no
sampler is provided, one must be provided using the
`set_default_sampler` function.
lagrange : optional (default 2)
Lagrange parameter to weight constraints (no edges within set)
versus objective (largest set possible).
sampler_args
Additional keyword parameters are passed to the sampler.
Returns
-------
clique_nodes : list
List of nodes that form a maximum clique, as
determined by the given sampler.
Notes
-----
Samplers by their nature may not return the optimal solution. This
function does not attempt to confirm the quality of the returned
sample.
References
----------
`Maximum Clique on Wikipedia <https://en.wikipedia.org/wiki/Maximum_clique(graph_theory)>`_
"""
return len(maximum_clique(G, sampler, lagrange, **sampler_args))
[docs]def is_clique(G, clique_nodes):
"""Determines whether the given nodes form a clique.
A clique is a subset of nodes of an undirected graph such that every two
distinct nodes in the clique are adjacent.
Parameters
----------
G : NetworkX graph
The graph on which to check the clique nodes.
clique_nodes : list
List of nodes that form a clique, as
determined by the given sampler.
Returns
-------
is_clique : bool
True if clique_nodes forms a clique.
Example
-------
This example checks two sets of nodes, both derived from a
single Chimera unit cell, for an independent set. The first set is
the horizontal tile's nodes; the second has nodes from the horizontal and
verical tiles.
>>> import dwave_networkx as dnx
>>> G = dnx.chimera_graph(1, 1, 4)
>>> dnx.is_clique(G, [0, 1, 2, 3])
False
>>> dnx.is_clique(G, [0, 4])
True
"""
for x in clique_nodes:
for y in clique_nodes:
if x != y:
if not(G.has_edge(x,y)):
return False
return True