In a (randomized) oblivious routing scheme the path chosen for a request between a source $s$ and a target $t$ is independent from the current traffic in the network. Hence, such a scheme consists of probability distributions over $s-t$ paths for every source-target pair $s,t$ in the network.
In a recent result \cite{R02} it was shown that for any undirected network there is an oblivious routing scheme that achieves a polylogarithmic competitive ratio with respect to congestion. Subsequently, Azar et al. \cite{ACF+03} gave a polynomial time algorithm that for a given network constructs the best oblivious routing scheme, i.e. the scheme that guarantees the best possible competitive ratio. Unfortunately, the latter result is based on the Ellipsoid algorithm; hence it is unpractical for large networks.
In this paper we present a combinatorial algorithm for constructing an oblivious routing scheme that guarantees a competitive ratio of $\O(\log^4n)$ for undirected networks. Furthermore, our approach yields a proof for the existence of an oblivious routing scheme with competitive ratio $\O(\log^3n)$, which is much simpler than the original proof from \cite{R02}.