Strongly Connected Spanning Subgraph for Almost Symmetric Networks

In the strongly connected spanning subgraph (SCSS) problem, the goal is to find a minimum weight spanning subgraph of a strongly connected directed graph that maintains the strong connectivity. In this paper, we consider the SCSS problem for two families of geometric directed graphs; t-spanners and symmetric disk graphs. Given a constant t ≥ 1, a directed graph G is a t-spanner of a set of points V if, for every two points u and v in V , there exists a directed path from u to v in G of length at most t · |uv|, where |uv| is the Euclidean distance between u and v. Given a set V of points in the plane such that each point u ∈ V has a radius ru, the symmetric disk graph of V is a directed graph G = (V,E), such that E = {(u, v) : |uv| ≤ ru and |uv| ≤ rv}. Thus, if there exists a directed edge (u, v), then (v, u) exists as well. We present 34 (t+ 1) and 3 2 approximation algorithms for the SCSS problem for t-spanners and for symmetric disk graphs, respectively. Actually, our approach achieves a 3 4 (t + 1)-approximation algorithm for all directed graphs satisfying the property that, for every two nodes u and v, the ratio between the shortest paths, from u to v and from v to u in the graph, is at most t.