  The vertex p-center problem consists of locating p facilities among a set of
M potential sites such that the maximum distance from any demand to its closest
located facility is minimized. The complete vertex p-center problem solves the
p-center problem for all p from 1 to the total number of sites, resulting in a
multi-objective trade-off curve between the number of facilities and the
service distance required to achieve full coverage. This trade-off provides a
reference to planners and decision-makers, enabling them to easily visualize
the consequences of choosing different coverage design criteria for the given
spatial configuration of the problem. We present two fast algorithms for
solving the complete p-center problem, one using the classical formulation but
trimming variables while still maintaining optimality, the other converting the
problem to a location set covering problem and solving for all distances in the
distance matrix. We also discuss scenarios where it makes sense to solve the
problem via brute-force enumeration. All methods result in significant
speed-ups, with the set covering method reducing computation times by many
orders of magnitude.
