Hungarian Assignment Problem

Hungarian Assignment Problem-40
When a number of agents and tasks is very large, a parallel algorithm with randomization can be applied.The problem of finding minimum weight maximum matching can be converted to finding a minimum weight perfect matching.Each edge (i,j), where i is in A and j is in T, has a weight .

When a number of agents and tasks is very large, a parallel algorithm with randomization can be applied.The problem of finding minimum weight maximum matching can be converted to finding a minimum weight perfect matching.

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In the above example, suppose that there are four taxis available, but still only three customers.It consists of finding, in a weighted bipartite graph, a matching in which the sum of weights of the edges is as large as possible.A common variant consists of finding a minimum-weight perfect matching.If the numbers of agents and tasks are equal, and the total cost of the assignment for all tasks is equal to the sum of the costs for each agent (or the sum of the costs for each task, which is the same thing in this case), then the problem is called the linear assignment problem.Commonly, when speaking of the assignment problem without any additional qualification, then the linear assignment problem is meant.The assignment problem can be solved by presenting it as a linear program.For convenience we will present the maximization problem.Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.It is required to perform all tasks by assigning exactly one agent to each task and exactly one task to each agent in such a way that the total cost of the assignment is minimized.Similar adjustments can be done in order to allow more tasks than agents, tasks to which multiple agents must be assigned (for instance, a group of more customers than will fit in one taxi), or maximizing profit rather than minimizing cost.The formal definition of the assignment problem (or linear assignment problem) is The problem is "linear" because the cost function to be optimized as well as all the constraints contain only linear terms.

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