Assignment problem

Common card · No. 310362

The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. 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 as many tasks as possible by assigning at most one agent to each task and at most one task to each agent, in such a way that the total cost of the assignment is minimized. Alternatively, describing the problem using graph theory: The assignment problem consists of finding, in a weighted bipartite graph, a matching of maximum size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment, and the graph-theoretic version is called minimum-cost perfect matching. Otherwise, it is called unbalanced assignment. If the total cost of the assignment for all tasks is equal...

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