  Locating source of diffusion in networks is crucial for controlling and
preventing epidemic risks. It has been studied under various probabilistic
models. In this paper, we study source location from a deterministic point of
view by modeling it as the minimum weighted doubly resolving set (DRS) problem,
which is a strengthening of the well-known metric dimension problem.
  Let $G$ be a vertex weighted undirected graph on $n$ vertices. A vertex
subset $S$ of $G$ is DRS of $G$ if for every pair of vertices $u,v$ in $G$,
there exist $x,y\in S$ such that the difference of distances (in terms of
number of edges) between $u$ and $x,y$ is not equal to the difference of
distances between $v$ and $x,y$. The minimum weighted DRS problem consists of
finding a DRS in $G$ with minimum total weight. We establish $\Theta(\ln n)$
approximability of the minimum DRS problem on general graphs for both weighted
and unweighted versions. This is the first work providing explicit
approximation lower and upper bounds for minimum (weighted) DRS problem, which
are nearly tight. Moreover, we design first known strongly polynomial time
algorithms for the minimum weighted DRS problem on general wheels and trees
with additional constant $k\ge0$ edges.
