Oblivious Routing for the Lp-norm

Gupta et al. [13] introduced a very general multicommodity flow problem in which the cost of a given flow solution on a graph G = (V, E) is calculated by first computing the link loads via a load-function ¿, that describes the load of a link as a function of the flow traversing the link, and then aggregating the individual link loads into a single number via an aggregation function agg:R^|E| ¿ R. In this paper we show the existence of an oblivious routing scheme with competitive ratio O(log n) and a lower bound of ¿(log n/log log n) for this model when the aggregation function agg is an L_p-norm. Our results can also be viewed as a generalization of the work on approximating metrics by a distribution over dominating tree metrics (see e.g. [4], [5], [8]) and the work on minimum congestion oblivious routing [20], [14], [21]. We provide a convex combination of trees such that routing according to the tree distribution approximately minimizes the L_p-norm of the link loads. The embedding techniques of Bartal [4], [5] and Fakcharoenphol et al. [8] can be viewed as solving this problem in the L₁-norm while the result of Racke [21] solves it for L_¿. We give a single proof that shows the existence of a good tree-based oblivious routing for any L_p-norm. For the Euclidean norm, we also show that it is possible to compute a tree-based oblivious routing scheme in polynomial time.