The assignment problem is a special case of linear programming problem used for allocating resources, mostly workforce, in an optimal way. It is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics. The goal of the assignment problem is to assign m resources, usually workers, to n tasks, usually jobs, on a one-to-one basis while minimizing assignment costs. All jobs must be performed by exactly one worker, and every worker must be assigned exclusively to one job. The problem arises because available resources such as men, machines, etc., have varying degrees of efficiency for performing different activities, and therefore, the cost, profit, or loss of performing the different activities is different.

The assignment problem can be formulated as a mathematical model, and the lpSolve package from R contains specific functions for solving linear programming assignment problems. The Hungarian method is a popular algorithm for solving the assignment problem. The problem can be extended to match more than two sets, resulting in the Multidimensional Assignment Problem (MAP).

In summary, the assignment problem is a special case of linear programming problem used for allocating resources in an optimal way. It involves assigning m resources to n tasks on a one-to-one basis while minimizing assignment costs. The problem can be formulated as a mathematical model and solved using algorithms such as the Hungarian method.