minimum_weight_full_matching(G, top_nodes=None, weight='weight')[source]

Returns the minimum weight full matching of the bipartite graph G.

Let \(G = ((U, V), E)\) be a complete weighted bipartite graph with real weights \(w : E \to \mathbb{R}\). This function then produces a maximum matching \(M \subseteq E\) which, since the graph is assumed to be complete, has cardinality

\[\lvert M \rvert = \min(\lvert U \rvert, \lvert V \rvert),\]

and which minimizes the sum of the weights of the edges included in the matching, \(\sum_{e \in M} w(e)\).

When \(\lvert U \rvert = \lvert V \rvert\), this is commonly referred to as a perfect matching; here, since we allow \(\lvert U \rvert\) and \(\lvert V \rvert\) to differ, we follow Karp 1 and refer to the matching as full.

  • G (NetworkX graph) – Undirected bipartite graph

  • top_nodes (container) – Container with all nodes in one bipartite node set. If not supplied it will be computed.

  • weight (string, optional (default=’weight’)) – The edge data key used to provide each value in the matrix.


matches – The matching is returned as a dictionary, matches, such that matches[v] == w if node v is matched to node w. Unmatched nodes do not occur as a key in matches.

Return type


  • ValueError – Raised if the input bipartite graph is not complete.

  • ImportError – Raised if SciPy is not available.


The problem of determining a minimum weight full matching is also known as the rectangular linear assignment problem. This implementation defers the calculation of the assignment to SciPy.



Richard Manning Karp: An algorithm to Solve the m x n Assignment Problem in Expected Time O(mn log n). Networks, 10(2):143–152, 1980.