Assignment Problem

Keywords and Synonyms

Weighted bipartite matching    

Problem Definition

Assume that a complete bipartite graph, ${ G(X,Y,X \times Y) }$, with weights w(x, y) assigned to every edge (x, y) is given. A matching M is a subset of edges so that no two edges in M have a common vertex. A perfect matching is one in which all the nodes are matched. Assume that ${ |X|=|Y|=n }$. The weighted matching problem is to find a matching with the greatest total weight, where ${ w(M)=\sum_{e \in M} w(e) }$. Since G is a complete bipartite graph, it has a perfect matching. An algorithm that solves the weighted matching problem is due to Kuhn [4] and Munkres [6]. Assume that all edge weights are nonnegative.

Key Results

Define a feasible vertex labeling ℓ as a mapping from the set of vertices in G to the reals, where

$$ \ell(x) + \ell(y) \geq w(x,y) \:. $$

Call ${ \ell(x) }$ the label of vertex x. It is easy to compute a feasible vertex labeling as follows:

$$ \forall y \in Y\quad \ell(y) = 0 $$