On the Competitiveness of Oblivious Routing: A Statistical View

Oblivious routing is a static algorithm for routing arbitrary user demands with the property that the competitive ratio, the proportion of the maximum congestion to the best possible congestion, is minimal. Oblivious routing turned out surprisingly efficient in this worst-case sense: in undirected graphs, we pay only a logarithmic performance penalty, and this penalty is usually smaller than 2 in directed graphs as well. However, compared to an optimal adaptive algorithm, which never causes congestion when subjected to a routable demand, oblivious routing surely has congestion. The open question is of how often is the network in a congested state. In this paper, we study two performance measures naturally arising in this context: the probability of congestion and the expected value of congestion. Our main result is the finding that, in certain directed graphs on n nodes, the probability of congestion approaches 1 in some undirected graphs, despite the competitive ratio being O(1).