Oblivious Routing in the L${}_p$-norm

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 $agg:{ℝ}^{\left|E\right|}\to ℝ$.

In this paper we show the existence of an oblivious routing scheme with competitive ratio $O(logn)$ and a lower bound of $\Omega (logn/loglogn)$ 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 $L_{\mathrm{\infty }}$. 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.