Setting
We are given a nonnegative n×n matrix, where the element in the i-th row and j-th column represents the cost of assigning the j-th job to the i-th worker. We have to find an assignment of the jobs to the workers that has minimum cost. If the goal is to find the assignment that yields the maximum cost, the problem can be altered to fit the setting by replacing each cost with the maximum cost subtracted by the cost.
The algorithm is easier to describe if we formulate the problem using a bipartite graph. We have a complete bipartite graph G=(S, T; E) with n worker vertices (S) and n job vertices (T), and each edge has a nonnegative cost c(i,j). We want to find a perfect matching with minimum cost.
Let us call a function a potential if for each . The value of potential y is . It can be seen that the cost of each perfect matching is at least the value of each potential. The Hungarian method finds a perfect matching and a potential with equal cost/value which proves the optimality of both. In fact it finds a perfect matching of tight edges: an edge ij is called tight for a potential y if . Let us denote the subgraph of tight edges by . The cost of a perfect matching in (if there is one) equals the value of y.
Read more about this topic: Hungarian Algorithm
Famous quotes containing the word setting:
“High from the summit of a craggy cliff,
Hung o’er the deep, such as amazing frowns
On utmost Kilda’s shore, whose lonely race
Resign the setting sun to Indian worlds,
The royal eagle draws his vigorous young”
—James Thomson (1700–1748)
“With wonderful art he grinds into paint for his picture all his moods and experiences, so that all his forces may be brought to the encounter. Apparently writing without a particular design or responsibility, setting down his soliloquies from time to time, taking advantage of all his humors, when at length the hour comes to declare himself, he puts down in plain English, without quotation marks, what he, Thomas Carlyle, is ready to defend in the face of the world.”
—Henry David Thoreau (1817–1862)
“We believe that Carlyle has, after all, more readers, and is better known to-day for this very originality of style, and that posterity will have reason to thank him for emancipating the language, in some measure, from the fetters which a merely conservative, aimless, and pedantic literary class had imposed upon it, and setting an example of greater freedom and naturalness.”
—Henry David Thoreau (1817–1862)