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 LIP the repeated queries and saving space, we describe the construction of a sparse subgraph Q subset of G, which contains approximately an 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 c-approximate shortest paths between all pairs of vertices with high probability. The number of edges in Q is 0(p(-c)n(1+2/c) log n) for edge-induced random subgraphs and O(p(-2c)n(1+2/c), log(2) n) for vertex-induced random subgraphs, where n is the number of vertices in G, p the sampling probability of edges/vertices, and C epsilon Z, c >= 3 is the desired approximation factor. (c) 2006 Wiley Periodicals, Inc.