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:
“When I consider the clouds stretched in stupendous masses across the sky, frowning with darkness or glowing with downy light, or gilded with the rays of the setting sun, like the battlements of a city in the heavens, their grandeur appears thrown away on the meanness of my employment; the drapery is altogether too rich for such poor acting. I am hardly worthy to be a suburban dweller outside those walls.”
—Henry David Thoreau (18171862)
“High from the summit of a craggy cliff,
Hung oer the deep, such as amazing frowns
On utmost Kildas shore, whose lonely race
Resign the setting sun to Indian worlds,
The royal eagle draws his vigorous young”
—James Thomson (17001748)
“The new sound-sphere is global. It ripples at great speed across languages, ideologies, frontiers and races.... The economics of this musical esperanto is staggering. Rock and pop breed concentric worlds of fashion, setting and life-style. Popular music has brought with it sociologies of private and public manner, of group solidarity. The politics of Eden come loud.”
—George Steiner (b. 1929)