Assignment Problem - Formal Mathematical Definition

Formal Mathematical Definition

The formal definition of the assignment problem (or linear assignment problem) is

Given two sets, A and T, of equal size, together with a weight function C : A × T → R. Find a bijection f : A → T such that the cost function:

is minimized.

Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as:

The problem is "linear" because the cost function to be optimized as well as all the constraints contain only linear terms.

The problem can be expressed as a standard linear program with the objective function

subject to the constraints

The variable represents the assignment of agent to task, taking value 1 if the assignment is done and 0 otherwise. This formulation allows also fractional variable values, but there is always an optimal solution where the variables take integer values. This is because the constraint matrix is totally unimodular. The first constraint requires that every agent is assigned to exactly one task, and the second constraint requires that every task is assigned exactly one agent.