The diameter of randomly perturbed digraphs and some applications

The central observation of this paper is that if εn random arcs are added to any n‐node strongly connected digraph with bounded degree then the resulting graph has diameter ??(lnn) with high probability. We apply this to smoothed analysis of algorithms and property testing. Smoothed Analysis: Recognizing strongly connected digraphs is a basic computational task in graph theory. Even for digraphs with bounded degree, it is NL‐complete. By XORing an arbitrary bounded degree digraph with a sparse random digraph R ～ ??_n,ε/n we obtain a “smoothed” instance. We show that, with high probability, a log‐space algorithm will correctly determine if a smoothed instance is strongly connected. We also show that if NL ?? almost‐L then no heuristic can recognize similarly perturbed instances of (s,t)‐connectivity. Property Testing: A digraph is called k‐linked if, for every choice of 2k distinct vertices s₁,…,s_k,t₁,…,t_k, the graph contains k vertex disjoint paths joining s_r to t_r for r = 1,…,k. Recognizing k‐linked digraphs is NP‐complete for k ≥ 2. We describe a polynomial time algorithm for bounded degree digraphs, which accepts k‐linked graphs with high probability, and rejects all graphs that are at least εn arcs away from being k‐linked. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007