Shortest-path metric approximation for random subgraphs

We consider graph optimization problems where the cost of a solution depends only on the shortest-path metric in the graph, such as Steiner Tree or Traveling Salesman. We study a scenario where such a problem needs to be solved repeatedly on random subgraphs of a given graph G. With the goal of speeding up the repeated queries and saving space, we describe the construction of a sparse subgraph Q ⊂ G which contains an approximately optimal solution for any such problem on a random subgraph of G, with high probability. More precisely, the subgraph Q has the property that after some vertices or edges are removed randomly, Q still contains capproximate shortest paths between all pairs of vertices with high probability. The number of edges in Q is O(pn log n) for edge-induced random subgraphs and O(pn log n) for vertex-induced random subgraphs, where n is the number of vertices in G, p the sampling probability of edges/vertices, and c ∈ Z, c ≥ 3 is the desired approximation factor.