Gupta et al. [GHR06] introduced a very general multi-commodity 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 \ell, 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:\mathbb{R}^{|E|}\rightarrow\mathbb{R}.
In this paper we show the existence of an oblivious routing scheme with competitive ratio O(\log n) and a lower bound of \Omega(\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. [Bar96,Bar98,FRT03]) and the work on minimum congestion oblivious routing [Rae02,HHR03,Rae08]. 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 [Bar96,Bar98] and Fakcharoenphol et al. [FRT03] can be viewed as solving this problem in the L_1-norm while the result of Räcke [Rae08] solves it for L_\infty. 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.