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 he 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 tighter lower bound 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 $O(log_2(NM))$ 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 solution 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 very recent exact methods by an order of magnitude. Moreover, it is the first one to solve large instances of size up to $N=M=1817$.

## Citation

ULB-ISRO preprint 2001/03. October 2001. ULB-ISRO- Service Optimisation Bld du Triomphe- CP210/01 B-1050 Brussels Belgium