The ArcGIS Location-Allocation does not use any integer programming to solve this complicated combinatorial problem. It uses heuristics. The LP approach cannot be used to solve lvery arge problems and can take too long to converge to a good solution.
Details about the algorithm are listed here:
Minimize Facilities is the same as Maximize Coverage but with the exception of the number of facilities to locate, which in this case is determined by the solver
The help on Minimize Facilities here:
So it is a set covering but what it does is starts with an estimate of the number of facilities needed and internally calls a variation of maximize coveraege to quickly and intelligently narrow down the number of minimum facilities needed and then finishes by running maximize coverage with the minimum facilities needed to populate the results.