New formulation and resolution method for the p-center problem

The p-Center problem consists in locating p facilities among a set of M possible locations and assigning N clients to them in order to minimize the maximum distance between a client and the facility to which it is allocated. We present a new integer linear programming formulation for this Min-Max problem with a polynomial number of variables and constraints, and show that its LP-relaxation provides a lower bound tighter than the classical one. Moreover, we show that an even better lower bound LB*, obtained by keeping the integrality restrictions on a subset of the variables, can be computed in polynomial time by solving at most O(log2(NM))linear programs, each made of N rows and M columns. We also show that, when the distances satisfy triangle inequalities, LB* is at least equal to half of the optimal value. Finally, we use LB* as a starting point in an exact solution method and report extensive computational results on test problems from the literature. For Euclidean instances, our method outperforms the runtime of other recent exact methods by an order of magnitude. Moreover, it is the first one to solve large instances of size up to N = M = 1817.