Let G=(V,E) be a graph with unit-length edges and nonnegative costs
assigned to its vertices. Being given a list of pairwise different vertices
S=(s1β,s2β,β¦,spβ), the {\em prioritized Voronoi diagram} of G with
respect to S is the partition of G in p subsets V1β,V2β,β¦,Vpβ so
that, for every i with 1β€iβ€p, a vertex v is in Viβ if and
only if siβ is a closest vertex to v in S and there is no closest vertex
to v in S within the subset {s1β,s2β,β¦,siβ1β}. For every i
with 1β€iβ€p, the {\em load} of vertex siβ equals the sum of the
costs of all vertices in Viβ. The load of S equals the maximum load of a
vertex in S. We study the problem of adding one more vertex v at the end of
S in order to minimize the load. This problem occurs in the context of
optimally locating a new service facility ({\it e.g.}, a school or a hospital)
while taking into account already existing facilities, and with the goal of
minimizing the maximum congestion at a site. There is a brute-force algorithm
for solving this problem in O(nm) time on n-vertex m-edge graphs.
We prove a matching time lower bound for the special case where m=n1+o(1)
and p=1, assuming the so called Hitting Set Conjecture of Abboud et al. On
the positive side, we present simple linear-time algorithms for this problem on
cliques, paths and cycles, and almost linear-time algorithms for trees, proper
interval graphs and (assuming p to be a constant) bounded-treewidth graphs