The diameter of randomly perturbed digraphs and some applications

Abstract

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 𝒪(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 ∼ 𝔻n,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,…,sk,t1,…,tk, the graph contains k vertex disjoint paths joining sr to tr 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