Hungarian Assignment Problem Protestant Reformation Essay

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.

In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks.One of the first such algorithms was the Hungarian algorithm, developed by Munkres.Other algorithms include adaptations of the primal simplex algorithm, and the auction algorithm.The assignment problem is a special case of the transportation problem, which is a special case of the minimum cost flow problem, which in turn is a special case of a linear program.While it is possible to solve any of these problems using the simplex algorithm, each specialization has more efficient algorithms designed to take advantage of its special structure.As shown by Mulmuley, Vazirani and Vazirani, the problem of minimum weight perfect matching is converted to finding minors in the adjacency matrix of a graph.Using the isolation lemma, a minimum weight perfect matching in a graph can be found with probability at least ½. Suppose that a taxi firm has three taxis (the agents) available, and three customers (the tasks) wishing to be picked up as soon as possible.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.Each edge (i,j), where i is in A and j is in T, has a weight .The goal is to find a maximum-weight perfect matching.

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