networkx.algorithms.clique.max_weight_clique

max_weight_clique(G, weight='weight')[source]

Find a maximum weight clique in G.

A clique in a graph is a set of nodes such that every two distinct nodes are adjacent. The weight of a clique is the sum of the weights of its nodes. A maximum weight clique of graph G is a clique C in G such that no clique in G has weight greater than the weight of C.

Parameters

  • G (NetworkX graph) – Undirected graph

  • weight (string or None, optional (default=’weight’)) – The node attribute that holds the integer value used as a weight. If None, then each node has weight 1.

Returns

  • clique (list) – the nodes of a maximum weight clique

  • weight (int) – the weight of a maximum weight clique

Notes

The implementation is recursive, and therefore it may run into recursion depth issues if G contains a clique whose number of nodes is close to the recursion depth limit.

At each search node, the algorithm greedily constructs a weighted independent set cover of part of the graph in order to find a small set of nodes on which to branch. The algorithm is very similar to the algorithm of Tavares et al. 1, other than the fact that the NetworkX version does not use bitsets. This style of algorithm for maximum weight clique (and maximum weight independent set, which is the same problem but on the complement graph) has a decades-long history. See Algorithm B of Warren and Hicks 2 and the references in that paper.

References

1

Tavares, W.A., Neto, M.B.C., Rodrigues, C.D., Michelon, P.: Um algoritmo de branch and bound para o problema da clique máxima ponderada. Proceedings of XLVII SBPO 1 (2015).

2

Warrent, Jeffrey S, Hicks, Illya V.: Combinatorial Branch-and-Bound for the Maximum Weight Independent Set Problem. Technical Report, Texas A&M University (2016).