maximum_independent_set(G, sampler=None, lagrange=2.0, **sampler_args)[source]

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.

  • G (NetworkX graph) – The graph on which to find a maximum cut independent set.
  • 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.

indep_nodes – List of nodes that form a maximum independent set, as determined by the given sampler.

Return type:



This example uses a sampler from dimod to find a maximum independent set for a graph of a Chimera unit cell created using the chimera_graph() function.

>>> import dimod
>>> sampler = dimod.SimulatedAnnealingSampler()
>>> G = dnx.chimera_graph(1, 1, 4)
>>> indep_nodes = dnx.maximum_independent_set(G, sampler)


Samplers by their nature may not return the optimal solution. This function does not attempt to confirm the quality of the returned sample.


Independent Set on Wikipedia

QUBO on Wikipedia

[AL]Lucas, A. (2014). Ising formulations of many NP problems. Frontiers in Physics, Volume 2, Article 5.