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.
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.
A common variant consists of finding a minimum-weight perfect matching.
It is a specialization of the maximum weight matching problem for bipartite graphs.
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.
The assignment problem is a fundamental combinatorial optimization problem.
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.