from dwave_networkx.utils import binary_quadratic_model_sampler
__all__ = ["maximum_independent_set", "is_independent_set"]
[docs]@binary_quadratic_model_sampler(1)
def maximum_independent_set(G, sampler=None, **sampler_args):
"""Returns an approximate maximum independent set.
Defines a QUBO with ground states corresponding to a
maximum independent set and uses the sampler to sample from
it.
An independent set is a set of nodes such that the subgraph
of G induced by these nodes contains no edges. A maximum
independent set is an independent set of largest possible size.
Parameters
----------
G : NetworkX graph
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.
sampler_args
Additional keyword parameters are passed to the sampler.
Returns
-------
indep_nodes : list
List of nodes that the form a maximum independent set, as
determined by the given sampler.
Examples
--------
>>> G = nx.path_graph(5)
>>> dnx.maximum_independent_set(G, sampler)
[0, 2, 4]
Notes
-----
Samplers by their nature may not return the optimal solution. This
function does not attempt to confirm the quality of the returned
sample.
https://en.wikipedia.org/wiki/Independent_set_(graph_theory)
https://en.wikipedia.org/wiki/Quadratic_unconstrained_binary_optimization
References
----------
.. [AL] Lucas, A. (2014). Ising formulations of many NP problems.
Frontiers in Physics, Volume 2, Article 5.
"""
# We assume that the sampler can handle an unstructured QUBO problem, so let's set one up.
# Let us define the largest independent set to be S.
# For each node n in the graph, we assign a boolean variable v_n, where v_n = 1 when n
# is in S and v_n = 0 otherwise.
# We call the matrix defining our QUBO problem Q.
# On the diagnonal, we assign the linear bias for each node to be -1. This means that each
# node is biased towards being in S
# On the off diagnonal, we assign the off-diagonal terms of Q to be 2. Thus, if both
# nodes are in S, the overall energy is increased by 2.
Q = {(node, node): -1 for node in G}
Q.update({edge: 2 for edge in G.edges})
# use the sampler to find low energy states
response = sampler.sample_qubo(Q, **sampler_args)
# we want the lowest energy sample
sample = next(iter(response))
# nodes that are spin up or true are exactly the ones in S.
return [node for node in sample if sample[node] > 0]
[docs]def is_independent_set(G, indep_nodes):
"""Determines whether the given nodes form an independent set.
An independent set is a set of nodes such that the subgraph
of G induced by these nodes contains no edges.
Parameters
----------
G : NetworkX graph
indep_nodes : list
List of nodes that the form a maximum independent set, as
determined by the given sampler.
Returns
-------
is_independent : bool
True if indep_nodes form an independent set.
"""
return not bool(G.subgraph(indep_nodes).edges)