Balancing graph Voronoi diagrams with one more vertex

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 $S=(s_1,s_2,\ldots,s_p)$, the {\em prioritized Voronoi diagram} of $G$ with respect to $S$ is the partition of $G$ in $p$ subsets $V_1,V_2,\ldots,V_p$ so that, for every $i$ with $1 \leq i \leq p$, a vertex $v$ is in $V_i$ if and only if $s_i$ is a closest vertex to $v$ in $S$ and there is no closest vertex to $v$ in $S$ within the subset $\{s_1,s_2,\ldots,s_{i-1}\}$. For every $i$ with $1 \leq i \leq p$, the {\em load} of vertex $s_i$ equals the sum of the costs of all vertices in $V_i$. 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 ${\cal O}(nm)$ time on $n$-vertex $m$-edge graphs. We prove a matching time lower bound for the special case where $m=n^{1+o(1)}$ 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.