Shortest‐path metric approximation for random subgraphs

Abstract

AbstractWe 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 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 O(p−cn1+2/c log n) for edge‐induced random subgraphs and O(p−2cn1+2/c log 2n) 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. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007