Linear bottleneck assignment problem


In combinatorial optimization, a field within mathematics, the linear bottleneck assignment problem is similar to the linear assignment problem.
In plain words the problem is stated as follows:
The term "" is explained by a common type of application of the problem, where the cost is the duration of the task performed by an agent. In this setting the "maximum cost" is "maximum duration", which is the bottleneck for the schedule of the overall job, to be minimized.

Formal definition

The formal definition of the bottleneck assignment problem is
Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as:

Mathematical programming formulation

subject to:

Asymptotics

Let denote the optimal objective function value for the problem with n agents and n tasks. If the costs are sampled from the uniform distribution on, then
and