Assignment Problem

Years and Authors of Summarized Original Work

• 1955; Kuhn

• 1957; Munkres

Problem Definition

Assume that a complete bipartite graph, G(X, Y, X × 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 $\vert X\vert =\vert Y \vert = n$. The weighted matching problem is to find a matching with the greatest total weight, where $w\left (M\right ) =\sum \limits _{e\in M}w\left (e\right )$. 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 non-negative.

Key Results

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

$$\displaystyle{\ell\left (x\right ) +\ell \left (y\right ) \geq w\left (x,y\right ).}$$

Call ℓ(x) the label of vertex x. It is...