An efficient benders decomposition for the p-median problem

The p-median problem is a classic discrete location problem with numerous applications. It aims to open p sites while minimizing the sum of the distances of each client to its nearest open site. We study a Benders decomposition of the most efficient formulation in the literature. We show that the Benders cuts can be separated in linear time. The Benders reformulation leads to a compact formulation for the p-median problem. We implement a two-phase Benders decomposition algorithm that outperforms state-of-the-art methods on benchmark instances by an order of magnitude and allows to exactly solve for the first time several instances among which are large TSP instances and BIRCH instances. We also show that our implementation easily applies to the uncapacitated facility location problem.