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The Hungarian method is a combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal–dual methods.It was developed and published in 1955 by Harold Kuhn, who gave it the name "Hungarian method" because the algorithm was largely based on the earlier works of two Hungarian mathematicians, Dénes Kőnig and Jenő Egerváry.
He described the Hungarian method for the assignment problem, but a paper by Carl Gustav Jacobi, published posthumously in 1890 in Latin, was later discovered that had described the Hungarian method a century before Kuhn. [1] [2]
In the balanced assignment problem, both parts of the bipartite graph have the same number of vertices, denoted by n. One of the first polynomial-time algorithms for balanced assignment was the Hungarian algorithm. It is a global algorithm – it is based on improving a matching along augmenting paths (alternating paths between unmatched vertices
This contribution was translated and published in 1955 by Harold W. Kuhn, [6] who also showed how to apply Kőnig's and Egerváry's method to solve the assignment problem; the resulting algorithm has since been known as the "Hungarian method". [7]
Hungarian algorithm unbalanced assignment problem example: Image title: Worked example of minimising costs by assigning tasks to an unequal number of workers using the Hungarian method, by CMG Lee. Width: 100%: Height: 100%
In the special case in which all the agents' budgets and all tasks' costs are equal to 1, this problem reduces to the assignment problem. When the costs and profits of all tasks do not vary between different agents, this problem reduces to the multiple knapsack problem. If there is a single agent, then, this problem reduces to the knapsack problem.
Some troops leave the battlefield injured. Others return from war with mental wounds. Yet many of the 2 million Iraq and Afghanistan veterans suffer from a condition the Defense Department refuses to acknowledge: Moral injury.
The formal definition of the bottleneck assignment problem is Given two sets, A and T, together with a weight function C : A × T → R. Find a bijection f : A → T such that the cost function: (, ()) is minimized.