*This may be very inefficient since, with n agents and n tasks, there are n! Many algorithms have been developed for solving the assignment problem in time bounded by a polynomial of n.One of the first such algorithms was the Hungarian algorithm, developed by Munkres.*

*This may be very inefficient since, with n agents and n tasks, there are n! Many algorithms have been developed for solving the assignment problem in time bounded by a polynomial of n.*One of the first such algorithms was the Hungarian algorithm, developed by Munkres.

The problem of finding minimum weight maximum matching can be converted to finding a minimum weight perfect matching.

A bipartite graph can be extended to a complete bipartite graph by adding artificial edges with large weights.

The solution to the assignment problem will be whichever combination of taxis and customers results in the least total cost.

However, the assignment problem can be made rather more flexible than it first appears.

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.

The firm prides itself on speedy pickups, so for each taxi the "cost" of picking up a particular customer will depend on the time taken for the taxi to reach the pickup point.

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.

When a number of agents and tasks is very large, a parallel algorithm with randomization can be applied.

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.

The assignment problem can be solved by presenting it as a linear program.

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