Approximation for the minimum cost doubly resolving set problem

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 cost doubly resolving set (DRS) problem, which is a strengthening of the well-known metric dimension problem.Let G be an undirected graph on n vertices, where each vertex has a nonnegative cost. A vertex subset S of G is a doubly resolving set (DRS) of G   if for every pair of vertices u,vu,v in G  , there exist x,y∈Sx,y∈S such that the difference of distances (in terms of number of edges) between u   and x,yx,y is not equal to the difference of distances between v   and x,yx,y. The minimum cost DRS problem consists of finding a DRS in G   with minimum total cost. We establish Θ(ln⁡n)Θ(ln⁡n) approximability of the minimum DRS problem on general graphs for both weighted and unweighted versions. This provides the first explicit lower and upper bounds on approximation for the minimum (cost) DRS, which are nearly tight. Moreover, we design the first known strongly polynomial time exact algorithms for the minimum cost DRS problem on general wheels and trees with additional constant k≥0k≥0 edges.