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However, we can solve it without the integrality constraints (i.e., drop the last constraint), using standard methods for solving continuous linear programs.
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.
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.
A common variant consists of finding a minimum-weight perfect matching.
Bottleneck Assignment Problem
It is a specialization of the maximum weight matching problem for bipartite graphs.
Without loss of generality we can assume that the cycle ends at vertex i1, so the last fractional variable in the cycle is .
So the number of edges in the cycle is 2m - it must be even since the graph is bipartite.
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.