# minimum_weight_full_matching#

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

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

Let $$G = ((U, V), E)$$ be a weighted bipartite graph with real weights $$w : E \to \mathbb{R}$$. This function then produces a matching $$M \subseteq E$$ with cardinality

$\lvert M \rvert = \min(\lvert U \rvert, \lvert V \rvert),$

which minimizes the sum of the weights of the edges included in the matching, $$\sum_{e \in M} w(e)$$, or raises an error if no such matching exists.

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.

Parameters:
GNetworkX graph

Undirected bipartite graph

top_nodescontainer

Container with all nodes in one bipartite node set. If not supplied it will be computed.

weightstring, optional (default=’weight’)

The edge data key used to provide each value in the matrix. If None, then each edge has weight 1.

Returns:
matchesdictionary

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.

Raises:
ValueError

Raised if no full matching exists.

ImportError

Raised if SciPy is not available.

Notes

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.

References

[1]

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.