We prove the existence of an oblivious routing scheme that is $mathrm{poly}(log n)$-competitive in terms of $(congestion + dilation)$, thus resolving a well-known question in oblivious routing. Concretely, consider an undirected network and a set of packets each with its own source and destination. The objective is to choose a path for each packet, from its source to its destination, so as to minimize $(congestion + dilation)$, defined as follows: The dilation is the maximum path hop-length, and the congestion is the maximum number of paths that include any single edge. The routing scheme obliviously and randomly selects a path for each packet independent of (the existence of) the other packets. Despite this obliviousness, the selected paths have $(congestion + dilation)$ within a $mathrm{poly}(log n)$ factor of the best possible value. More precisely, for any integer hop-bound $h$, this oblivious routing scheme selects paths of length at most $h cdot mathrm{poly}(log n)$ and is $mathrm{poly}(log n)$-competitive in terms of $congestion$ in comparison to the best possible $congestion$ achievable via paths of length at most $h$ hops. These paths can be sampled in polynomial time. This result can be viewed as an analogue of the celebrated oblivious routing results of R{a}cke [FOCS 2002, STOC 2008], which are $O(log n)$-competitive in terms of $congestion$, but are not competitive in terms of $dilation$.
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